About the Landau-Fermi-Dirac Equation With Moderately Soft Potentials

Abstract

We present some essential properties of solutions to the homogeneous Landau-Fermi-Dirac equation for moderately soft potentials. Uniform in time estimates for statistical moments, $L^{p}$-norm generation and Sobolev regularity are shown using a combination of techniques that include recent developments concerning level set analysis in the spirit of De Giorgi and refined entropy-entropy dissipation functional inequalities for the Landau collision operator which are extended to the case in question here. As a consequence of the analysis, we prove algebraic relaxation of non degenerate distributions towards the Fermi-Dirac statistics under a weak non saturation condition for the initial datum. All quantitative estimates are uniform with respect to the quantum parameter. They therefore also hold for the classical limit, that is, the Landau equation.

1 Introduction

1.1 Setting of the problem

In the ensuing pages we study the essential properties of a dilute gas satisfying Pauli’s exclusion principle in the Landau’s grazing limit regime. More specifically, we study the Landau-Fermi-Dirac (LFD) equation in the homogeneous setting for moderately soft potential interactions described as

$$\begin{aligned} \partial _{t} f(t,v) = {\mathcal {Q}}(f)(t,v),\qquad (t,v)\in (0,\infty )\times {\mathbb {R}}^{3}\,, \qquad f(0)=f_{\mathrm {in}}\,, \end{aligned}$$

(1.1)

where the collision operator ${\mathcal {Q}}$ is given by a modification of the Landau operator which includes Pauli’s exclusion principle. This is defined as

$$\begin{aligned}&{\mathcal {Q}}(f)(v)= {\nabla }_v \cdot \int _{{\mathbb {R}}^3} \Psi (v-v_{*}) \, \Pi (v-v_{*})\nonumber \\&\Big \{ f_{*}(1- \vec {\varvec{\varepsilon }} f_{*}) \nabla f - f (1- \vec {\varvec{\varepsilon }} f) {\nabla f}_*\Big \} \, \mathrm {d}v_{*}\, , \end{aligned}$$

(1.2)

with the usual shorthand $f:= f(v)$, $f_* := f(v_*)$, and

$$\begin{aligned} \Psi (v-v_{*})=|v-v_{*}|^{\gamma +2}, \qquad \Pi (z)=\mathrm {Id} - \frac{z \otimes z}{|z|^{2}}. \end{aligned}$$

The Pauli exclusion principle implies that a solution to (1.11.10) must a priori satisfy the bound

$$\begin{aligned} 0\leqslant f(t,v)\leqslant \varvec{\varepsilon }^{-1}, \end{aligned}$$

where the quantum parameter

$$\begin{aligned} \varvec{\varepsilon }:= \frac{(2\pi \hslash )^{3}}{m^{3}\beta } >0 \end{aligned}$$

depends on the reduced Planck constant $\hslash \approx 1.054\times 10^{-34} \mathrm {m}^{2}\mathrm {kg\,s}^{-1}$, the mass m and the statistical weight $\beta$ of the particles species; see [1, Chapter 17]. In the case of electrons $\varvec{\varepsilon }\approx 1.93\times 10^{-10}\ll 1$. The parameter $\varvec{\varepsilon }$ quantifies the quantum effects of the model. The case $\varvec{\varepsilon }=0$ corresponds to the classical Landau equation.

In this paper, we are interested in moderately soft potentials, corresponding to the case when $\gamma \in (-2,0)$. The main original features of this paper are:

• It is the first systematic study of the LFD equation for moderately soft potentials, which are a class of potentials essentially closer to the most relevant case of Coulomb interactions than the recently studied hard potentials case, see [2].

• Pointwise bounds are obtained thanks to a variant of the De Giorgi method, which leads to an elegant proof in which no high-order derivatives are manipulated. Such $L^{\infty }$-estimates are actually independent of the quantum parameter $\varvec{\varepsilon }$ and yield the following pointwise lower bound

  $$\begin{aligned} \inf _{v \in {\mathbb {R}}^{3}}\left( 1-\varvec{\varepsilon }f(t,v)\right) \geqslant \kappa _{0} >0\,, \qquad \forall t \geqslant 1, \end{aligned}$$

  (1.3)

  which plays a fundamental role in the long-time behaviour analysis.

• Stretched exponential decay towards equilibrium is recovered thanks to a careful analysis of the constants pertaining to the moments bounds and to a complex interpolation procedure involving a nonstandard Gronwall-like lemma; we point out that, for soft potentials, exponential decay is not expected.

• All estimates are uniform with respect to the quantum parameter (lying in a range fully determined by properties of the initial datum such as statistical moments and entropy), so that the statements and proofs also hold for the Landau equation with moderately soft potentials. This provides a new approach for classical and novel results concerning this equation, in particular related to the long time behaviour.

Concerning the smallness of the parameter in the above point, let us make clear to the reader that our approach does not resort to any kind of perturbation argument. The smallness of $\varvec{\varepsilon }$ has to be interpreted rather as a saturation condition since we need to ensure that $\varvec{\varepsilon }$ lies in some physical range for which the above (1.3) holds. In particular, this restriction on the range of parameters will be needed here only for the results regarding convergence towards equilibrium.

Before going further in the description of our results and the related literature on the field, let us comment a little bit on the terminology we adopt and the physical relevance of our results. As far as the classical Landau equation is concerned, the physically relevant potential interaction is the one associated to Coulomb interactions, corresponding to the choice $\gamma =-3$. This is well-documented in the plasma physics literature, and has been observed for instance in [3] in the context of the derivation of the Landau equation from a N particles system. Under such a premise, it may appear artificial to choose the range of parameters $\gamma \in (-2,0)$ for our analysis and to refer to it as “moderately soft potentials”. The choice of this terminology is motivated by the fact that the Landau-Fermi-Dirac equation (1.11.10) can be derived at the formal level from the Boltzmann-Fermi-Dirac equation in the grazing collision limit (see [1, 4, 5] for details in the classical case) and, as such, inherits the terminology in use for the Boltzmann Equation [6]. Besides the terminology, the choice of the range of parameters $-2< \gamma <0$ corresponds to a case in which a well understood theory exists for global, non-perturbative strong solutions of the classical (that is, when $\varvec{\varepsilon }=0$) Landau equation (such a theory exists in fact when $-2 \leqslant \gamma \leqslant 1$, cf. [7, 8]). For $\gamma < -2$, only weak (or H-) solutions are known to exist [9]. Since the main bounds presented in this work are uniform with respect to $\varvec{\varepsilon }$, we expect that they cannot easily be extended to the case when $\gamma < -2$, at least using the same methods. Possible extensions of the De Giorgi approach to derive pointwise bounds will nevertheless be the object of future work by the authors, for models related to the Landau equation. We finally mention the recent work [10] which deals with (1.11.10) for $\gamma =-3$ in a framework different from ours since the work [10] is dealing with some fixed $\varvec{\varepsilon }$ and is not concerned with uniform bounds with respect to $\varvec{\varepsilon }.$

1.2 Thermal equilibrium

The relevant steady state of the LFD equation is the so-called Fermi-Dirac statistics.

Definition 1.1

(Fermi-Dirac statistics) Given $\varrho>0, u\in {\mathbb {R}}^3, \theta >0$ satisfying

$$\begin{aligned} 5\theta >\left( \frac{3\varvec{\varepsilon }\varrho }{4\pi }\right) ^{\frac{2}{3}}, \end{aligned}$$

(1.4)

we denote by ${\mathcal {M}}_{\varvec{\varepsilon }}$ the unique Fermi-Dirac statistics (see [11,  Proposition 3] for the proof of existence and uniqueness of such a function)

$$\begin{aligned} {\mathcal {M}}_{\varvec{\varepsilon }}(v)=\frac{a_{\varvec{\varepsilon }}\exp (-b_{\varvec{\varepsilon }}|v-u|^{2})}{1+\varvec{\varepsilon }\,a_{\varvec{\varepsilon }}\exp (-b_{\varvec{\varepsilon }}|v-u|^{2})}=: \frac{M_{\varvec{\varepsilon }}}{1+\varvec{\varepsilon }\,M_{\varvec{\varepsilon }}}, \end{aligned}$$

(1.5)

with $a_{\varvec{\varepsilon }},$ $b_{\varvec{\varepsilon }}$ defined in such a way that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}{\mathcal {M}}_{\varvec{\varepsilon }}(v)\left( \begin{array}{c}1\\ v \\ |v-u|^{2} \end{array}\right) \, \mathrm {d}v =\left( \begin{array}{c}\varrho \\ \varrho \,u \\ 3\varrho \,\theta \end{array}\right) \,. \end{aligned}$$

Note that $M_{\varvec{\varepsilon }}$ is here a suitable Maxwellian distribution that allows us to recover in the classical limit $\varvec{\varepsilon }\rightarrow 0$ the Maxwellian equilibrium.

Besides the Fermi-Dirac statistics (1.5), the distribution

$$\begin{aligned} F_{\varvec{\varepsilon }}(v)={\left\{ \begin{array}{ll} \varvec{\varepsilon }^{-1} &{}\quad \text { if } |{v-u}|\leqslant \left( \dfrac{3\varrho \,\varvec{\varepsilon }}{|{\mathbb {S}}^{2}|} \right) ^{\frac{1}{3}}, \\ 0 &{}\quad \text { if } |{v-u}|> \left( \dfrac{3\varrho \,\varvec{\varepsilon }}{|{\mathbb {S}}^{2}|}\right) ^{\frac{1}{3}}\end{array}\right. } \end{aligned}$$

(1.6)

can be a stationary state with prescribed mass $\varrho =\int _{{\mathbb {R}}^{3}}F_{\varvec{\varepsilon }}(v)\mathrm {d}v$ (where $|{\mathbb {S}}^{2}|=4\pi$ is the volume of the unit sphere). Such a degenerate state, referred to as a saturated Fermi-Dirac stationary state, can occur for very cold gases (with an explicit condition on the gas temperature). For such saturated states, the condition

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}F_{\varvec{\varepsilon }}(v)\left( \begin{array}{c}1\\ v \\ |v-u|^{2} \end{array}\right) \, \mathrm {d}v =\left( \begin{array}{c}\varrho \\ \varrho \,u \\ 3\varrho \,\theta \end{array}\right) \, \end{aligned}$$

makes the inequality (1.4) an identity which enforces

$$\begin{aligned} \varvec{\varepsilon }= \varvec{\varepsilon }_{\text {sat}}:=\frac{4\pi \,(5\,\theta )^{\frac{3}{2}}}{3\varrho }\,. \end{aligned}$$

The fact that an initial distribution close to such degenerate state makes $1-\varvec{\varepsilon }f$ arbitrarily small in non negligible sets affects the diffusion mechanism and the regularisation process induced by the parabolic nature of (1.11.10). As such, the existence of such saturated states impacts the gas relaxation towards the corresponding Fermi-Dirac statistics in a close-to-saturation situation. It was shown in reference [2] that, for hard potentials, explicit exponential relaxation rates exist when $\varvec{\varepsilon }\in (0,c\,\varvec{\varepsilon }_{\text {sat}})$ for some universal $c\in (0,1)$. One of the central results of this work is the proof of an analogous statement for moderately soft potentials (with algebraic rates). Proving explicit relaxation rates for $c=1$ remains an open problem for any potential.

1.3 Notations

For $s \in {\mathbb {R}}$ and $p\geqslant 1$, we define the Lebesgue space $L^{p}_{s}({\mathbb {R}}^3)$ through the norm

$$\begin{aligned}&\displaystyle \Vert f\Vert _{L^p_{s}} := \left( \int _{{\mathbb {R}}^3} \big |f(v)\big |^p \, \langle v\rangle ^s \, \mathrm {d}v\right) ^{\frac{1}{p}},\\&\qquad L^{p}_{s}({\mathbb {R}}^3) :=\Big \{f\,:{\mathbb {R}}^{3} \rightarrow {\mathbb {R}}\;;\,\Vert f\Vert _{L^{p}_{s}} < \infty \Big \}\, , \end{aligned}$$

where $\langle v\rangle :=\sqrt{1+|v|^{2}}$, $v\in {\mathbb {R}}^{3}.$ More generally, for any weight function $\varpi \,:\,{\mathbb {R}}^{3} \rightarrow {\mathbb {R}}^{+}$, we define, for any $p \geqslant 1$,

$$\begin{aligned} L^{p}(\varpi ) :=\Big \{f\,:\,{\mathbb {R}}^{3} \rightarrow {\mathbb {R}}\;;\,\Vert f\Vert _{L^{p}(\varpi )}^{p}:=\int _{{\mathbb {R}}^{3}}\big |f\big |^{p}\,\varpi \,\mathrm {d}v < \infty \Big \}\,. \end{aligned}$$

With this notation, one can write for example $L^{p}_{s}({\mathbb {R}}^{3})=L^{p}\big (\langle \cdot \rangle ^{s}\big )$, for $p \geqslant 1,\,s \geqslant 0$. We define the weighted Sobolev spaces by

$$\begin{aligned} W^{k,p}_{s}({\mathbb {R}}^3) :=\Big \{f \in L^{p}_{s}({\mathbb {R}}^3)\;;\;\partial _{v}^{\beta }f \in L^{p}_{s}({\mathbb {R}}^3) \,\;\forall \, |\beta | \leqslant k\Big \}\,,\quad \text {with}\quad k \in {\mathbb {N}}\,, \end{aligned}$$

with the standard norm

$$\begin{aligned} \Vert f\Vert _{W^{k,p}_{s}} := \bigg ( \sum _{0\leqslant |\beta | \leqslant k} \int _{{\mathbb {R}}^3} \big | \partial ^{\beta }_v f(v)\big |^p\, \langle v\rangle ^s \, \mathrm {d}v\bigg )^{\frac{1}{p}}, \end{aligned}$$

where $\beta =(i_1,i_2,i_3)\in {\mathbb {N}}^3$, $|\beta |=i_1+i_2+i_3$ and $\partial ^{\beta }_v f =\partial _1^{i_1}\partial _2^{i_2}\partial _3^{i_3} f$. For $p=2$, we will simply write $H^{k}_{s}({\mathbb {R}}^{3})=W^{k,2}_{s}({\mathbb {R}}^{3})$, $k \in {\mathbb {N}}$, $s \geqslant 0$. An additional important shorthand that will be used when specifically referring to moments and weighted $L^{2}$-norm of solutions is defined in the following:

Definition 1.2

Given a nonnegative measurable map** $g\,:\,{\mathbb {R}}^{3}\rightarrow {{\mathbb {R}}^+}$, we introduce for any $s \in {\mathbb {R}}$,

and

Moreover, if $f=f(t,v)$ is a (weak) solution to (1.11.10), we simply write

and .

1.4 Weak solutions for the moderately soft potential case $\gamma \in (-2,0)$

In the sequel we perform the calculations in the following functional framework:

Definition 1.3

Fix $\varvec{\varepsilon }_{0}>0$ and a nonnegative $f_{\mathrm {in}}\in L^{1}_{2}({\mathbb {R}}^{3})$ satisfying

$$\begin{aligned} 0<\Vert f_{\mathrm {in}}\Vert _{L^{\infty }} =:\varvec{\varepsilon }_{0}^{-1}<\infty \qquad \text { and } \qquad {\mathcal {S}}_{\varvec{\varepsilon }_{0}}(f_{\mathrm {in}})>0, \qquad |H(f_{\mathrm {in}})| < \infty \,, \end{aligned}$$

(1.7)

where ${\mathcal {S}}_{\varvec{\varepsilon }_{0}}(f_{\mathrm{in}})$ denotes the Landau-Fermi-Dirac entropy while $H(f_{\mathrm{in}})$ is the classical Boltzmann entropy (see Section 2.1 for precise definition).

For any $\varvec{\varepsilon }\in [0,\varvec{\varepsilon }_{0}]$, we say that $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$ if $f\in L^{1}_{2}({\mathbb {R}}^{3})$ satisfies $0\leqslant f\leqslant \varvec{\varepsilon }^{-1}$ and

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}f(v)\left( \begin{array}{c}1 \\ v \\ |v|^{2}\end{array}\right) \mathrm {d}v=\int _{{\mathbb {R}}^{3}}f_{\mathrm {in}}(v)\left( \begin{array}{c}1 \\ v \\ |v|^{2}\end{array}\right) \mathrm {d}v=\left( \begin{array}{c}\varrho \\ \varrho u \\ 3\varrho \theta +\varrho |u|^{2}\end{array}\right) , \end{aligned}$$

(1.8)

and ${\mathcal {S}}_{\varvec{\varepsilon }}(f) \geqslant {\mathcal {S}}_{\varvec{\varepsilon }}(f_{\mathrm {in}}).$

By a simple scaling argument, there is no loss in generality in assuming that

$$\begin{aligned} \varrho =\theta =1, \qquad u=0. \end{aligned}$$

(1.9)

This assumption will be made throughout the manuscript and ${\mathcal {M}}_{\varvec{\varepsilon }}$ will always denote the Fermi-Dirac statistics corresponding to this normalisation.

It is important to clarify the role of the class ${\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm{in}})$ in the sequel of the paper as well as that of $\varvec{\varepsilon }_{0}$. In all the subsequent results, the parameter $\varvec{\varepsilon }_{0} >0$ is fixed and $f_{\mathrm{in}}$ satisfying (1.7) is chosen. Then, in several results, we will consider a smaller threshold parameter, say $\varvec{\varepsilon }_{\star } \in (0,\varvec{\varepsilon }_{0}]$, and solutions $f=f(t,v)$ to (1.11.10) for all $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star }]$. Such solutions will belong to the class ${\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm{in}})$ and properties of such solutions as well as various bounds for them will be derived uniformly with respect to $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star }].$ We also wish to point out that, even though our analysis is performed in the physical space ${\mathbb {R}}^{3}$, there is no obstruction to reformulate all our results by considering $v \in {\mathbb {R}}^{d}$ with $d \geqslant 2$. The main tools used in our analysis (Sobolev embedding, Nash inequality, etc.) would still apply to this case, yielding only in a change of some of the threshold values $\varvec{\varepsilon }_{0},\varvec{\varepsilon }_{\star }$ appearing in the paper. For the clarity of exposition, we restrict ourselves to the case $d=3$.

We adopt the notations of [2], namely,

$$\begin{aligned} \left\{ \begin{aligned} a(z)&= \left( a_{i,j}(z)\right) _{i,j} \quad \text{ with } \quad a_{i,j}(z) = |z|^{\gamma +2} \,\left( \delta _{i,j} -\frac{z_i z_j}{|z|^2} \right) ,\\ b_i(z)&= \sum _k \partial _k a_{i,k}(z) = -2 \,z_i \, |z|^\gamma , \\ c(z)&= \sum _{k,l} \partial ^2_{kl} a_{k,l}(z) = -2 \,(\gamma +3) \, |z|^\gamma . \\ \end{aligned}\right. \end{aligned}$$

For any $f \in L^{1}_{2+\gamma }({\mathbb {R}}^{3})$, we define then the matrix-valued map**s $\varvec{\sigma }[f]$ and $\varvec{\Sigma }[f]$ given by

$$\begin{aligned} \varvec{\sigma }[f]=\big (\varvec{\sigma }_{ij}[f]\big )_{ij}:=\big (a_{ij}*f\big )_{ij}, \qquad \qquad \varvec{\Sigma }[f]=\varvec{\sigma }[f(1-\varvec{\varepsilon }\,f)]. \end{aligned}$$

In the same way, we set $\varvec{b}[f]\,:\,v \in {\mathbb {R}}^{3} \mapsto \varvec{b}[f](v) \in {\mathbb {R}}^{3}$ given by

$$\begin{aligned} \varvec{b}_{i}[f](v)=\big (b_{i} *f\big )(v), \qquad \forall \, v \in {\mathbb {R}}^{3},\qquad i=1,2,3. \end{aligned}$$

We also introduce

$$\begin{aligned} \varvec{B}[f]=\varvec{b}[f(1-\varvec{\varepsilon }\,f)], \qquad \text { and } \qquad \varvec{c}_{\gamma }[f]=c *f. \end{aligned}$$

We emphasise the dependency with respect to the parameter $\gamma$ in $\varvec{c}_{\gamma }[f]$ since, in several places, we apply the same definition with $\gamma +1$ replacing $\gamma$.

With these notations, the LFD equation can then be written alternatively under the form

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t} f = \nabla \cdot \big (\,\varvec{\Sigma }[f]\, \nabla f - \varvec{b}[f]\, f(1-\varvec{\varepsilon }f)\big ) , \\ f(t=0)= f_{\mathrm {in}}\,. \end{array}\right. \end{aligned}$$

(1.10)

Definition 1.4

Consider a non trivial initial datum $f_{\mathrm {in}} \in L^{1}_{2}({\mathbb {R}}^{3})$ satisfying (1.7)–(1.9) with $\varvec{\varepsilon }_{0} >0$ and let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$. A weak solution to the LFD Equation (1.11.10) is a function $f\,:\,{\mathbb {R}}^{+}\times {\mathbb {R}}^{3}\rightarrow {\mathbb {R}}^{+}$ satisfying the following conditions:

• (i) $f \in L^{\infty }({\mathbb {R}}^{+};L^{1}_{2}({\mathbb {R}}^{3})) \bigcap {\mathscr {C}}({\mathbb {R}}^{+},{\mathscr {D}}'({\mathbb {R}}^{3}))$,

• (ii) $f(t) \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$ for any $t \geqslant 0$ and $f(0)=f_{\mathrm {in}}$,

• (iii) The map** $t\mapsto {\mathcal {S}}_{\varvec{\varepsilon }}(f(t))$ is non-decreasing,

• (iv) For any $\varphi =\varphi (t,v) \in {\mathscr {C}}_{c}^{2}([0,T)\times {\mathbb {R}}^{3})$,

  $$\begin{aligned}&-\int _{0}^{T}\mathrm {d}t\int _{{\mathbb {R}}^{3}}f(t,v)\partial _{t}\varphi (t,v)\, \mathrm {d}v-\int _{{\mathbb {R}}^{3}}f_{\mathrm {in}}(v)\varphi (0,v)\mathrm {d}v\nonumber \\&\quad =\int _{0}^{T}\mathrm {d}t \int _{{\mathbb {R}}^{3}}\sum _{i,j}{\vec {\Sigma }}_{i,j}[f(t)]f(t,v) \partial ^{2}_{v_{i},v_{j}}\varphi (t,v)\mathrm {d}v\nonumber \\&\qquad + \sum _{i=1}^{3}\int _{0}^{T}\,\mathrm {d}t\int _{{\mathbb {R}}^{6}}f(t,v)f(t,w)(1-{\vec {\varvec{\varepsilon }}} f(t,w)){\vec {b}}_{i}(v-w)\nonumber \\&\qquad \left[ \partial _{v_{i}}\varphi (t,v) -\partial _{w_{i}}\varphi (t,w)\right] \,\mathrm {d}v\,\mathrm {d}w. \end{aligned}$$

  (1.11)

Notice that, since $f(t) \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm{in}})$, one has in particular $0 \leqslant f(t) \leqslant \varvec{\varepsilon }^{-1}$ for any $t \geqslant 0$. Since $\varphi$ has compact support together with its derivatives, all the terms in (1.11) are well defined.

1.5 Main results

As mentioned, we study the existence, uniqueness, smoothness, large velocity and large time behavior of solutions to the spatially homogeneous Landau-Fermi-Dirac Equation (1.11.10) with moderately soft potentials. We now present our main results and insist that all estimates provided are uniform in the vanishing limit of the quantum parameter $\varvec{\varepsilon }$.

We start with a result regarding the existence of weak solutions.

Theorem 1.5

Let $\gamma \in (-2,0]$. Consider an initial datum $f_{\mathrm {in}}\in L^1_{s_0}({\mathbb {R}}^3)$ for some $s_0>2$ satisfying (1.7)–(1.9) with $\varvec{\varepsilon }_{0} >0$. Then, for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ there exists a weak solution f to (1.11.10) and one has $f\in L^\infty _{\mathrm {loc}}({\mathbb {R}}_+,L^1_{s_0}({\mathbb {R}}^3))$.

The proof of this existence result can be found in Appendix B. It follows the same lines as the proof of the analogous theorem in the hard potential case in [12]. We recall that for the classical Landau equation (that is for $\varvec{\varepsilon }=0$) the theory of existence for the case when $\gamma <-2$ (very soft potentials) is substantially different from the case $\gamma > -2$ (moderately soft potentials) [8, 9]; we do not investigate the LFD equation with very soft potentials in this paper.

We now turn to a result of smoothness which holds uniformly with respect to $\varvec{\varepsilon }$, for any given time interval [0, T], with $T>0$. Uniformity with respect to T is not obtained at this level, and is considered only in next result.

Theorem 1.6

Let $\gamma \in (-2,0)$. Consider an initial datum $f_{\mathrm {in}}\in L^1_{s}({\mathbb {R}}^3) \cap L^{q_0}({\mathbb {R}}^3)$ for all $s \geqslant 0$ and some $q_0\geqslant 2$, satisfying (1.7)–(1.9) with $\varvec{\varepsilon }_{0} >0$. Then, for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$, any weak solution to Equation (1.11.10) constructed in Theorem 1.5 lies in $L^{\infty }([0,T]; L^q_s({\mathbb {R}}^3))$ for all $s\geqslant 0$, $q \in [1,q_{0})$ and $T>0$.

Moreover if the initial datum $f_{\mathrm {in}}$ also lies in $W^{1,p}_s({\mathbb {R}}^3)$ for all $s\geqslant 0$ and all $p \in [1, \infty )$, any weak solution constructed in Theorem 1.5 is unique and lies in $L^{\infty }([0,T]; W^{1,p}_s({\mathbb {R}}^3)) \cap {L^2}([0,T]; H^{2}_s({\mathbb {R}}^3))$ for all $s\geqslant 0$, $p \in [1, \infty )$ and $T>0$, as well as in ${\mathscr {C}}^{0,\alpha }([0,T] \times {\mathbb {R}}^3)$ for some $\alpha \in (0,1)$ and all $T>0$. Finally, all the norms of f in the spaces described in this Theorem are uniform with respect to $\varvec{\varepsilon }\in [0,\varvec{\varepsilon }_0]$ and depend on the $W^{1,p}_s({\mathbb {R}}^3)$ norms of $f_{\mathrm{in}}$ as well as $H(f_{\mathrm{in}})$.

The fact that the solution $f=f(t,v)$ belongs to ${\mathscr {C}}^{0,\alpha }([0,T] \times {\mathbb {R}}^{3})$ can be used to show that f is in fact a classical solution. The proof of this result of propagation of regularity can be found in Appendix A, see in particular Corollary A.8. It follows the methods used in [8] and [9]. Notice that stability (for finite intervals of time) and consequently uniqueness can be investigated thanks to the study of smoothness (for sufficiently smooth initial data).

It can be improved in many directions: The assumptions on initial data can be changed (cf. the various propositions in Appendix A); Appearance of regularity can be shown (this can also be seen in the various propositions in Appendix A); The dependence w.r.t. time of the estimates can be obtained explicitly (and involves only powers and no exponentials, since Gronwall’s lemma is not used), we refer to next theorem for the use of the large time behavior for obtaining uniformity w.r.t. time when (polynomial) moments of sufficient order are initially finite. Note that stretched exponential moments can be considered instead of algebraic moments, as is done in Section 6.

Concerning the long-time behaviour of the solution to (1.11.10), the main result of this work can be summarised in the following theorem:

Theorem 1.7

Assume that $\gamma \in \left( -2,0\right)$ and consider a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) with $\varvec{\varepsilon }_{0} >0$, with moreover $f_{\mathrm{in}} \in L^{1}_{s}({\mathbb {R}}^{3})$ with $s > 14+6|\gamma |.$ Then, there exists $\varvec{\varepsilon }_{\star } \in (0,\varvec{\varepsilon }_{0}]$ depending only on $f_{\mathrm {in}}$ through its $L^{1}_{s}$-norm such that for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star }]$, any nonnegative weak solution $f:= f(t,v)$ to (1.11.10) constructed in Theorem 1.5 satisfies

• (1) No Saturation : 

  $$\begin{aligned} \kappa _{0} := 1 - \varvec{\varepsilon }\,\sup _{t\geqslant 1}\Vert f(t) \Vert _{\infty }>0. \end{aligned}$$

• (2) Algebraic Relaxation :  there exists $C>0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ and s such that

  $$\begin{aligned} {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}) \leqslant C\,\left( 1+t\right) ^{ {-\frac{s-8-6|\gamma |}{2|\gamma |}}}\,, \qquad t \geqslant 1\,, \end{aligned}$$

  which implies, in particular, that

  $$\begin{aligned} \left\| f(t)-{\mathcal {M}}_{\varvec{\varepsilon }}\right\| _{L^{1}} \leqslant \sqrt{2C}\,\left( 1+t\right) ^{{-\frac{s-8-6|\gamma |}{4|\gamma |}}}\,, \qquad t \geqslant 1. \end{aligned}$$

Finally, if

$$\begin{aligned} f_{\mathrm{in}} \in L^{1}_{r}({\mathbb {R}}^{3}) \qquad \text { with } \quad r >\max \left( {2s + 8 + 2|\gamma |},\frac{s^{2}}{s-2|\gamma |}\right) , \end{aligned}$$

then there exists a constant $C(\gamma ,s,f_{\mathrm {in}})$ depending on $H(f_{\mathrm{in}})$, s, $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{r}}$ such that, for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star })$

$$\begin{aligned} \sup _{t\geqslant 1}\varvec{E}_{s}(t) + \sup _{t\geqslant 1}\Vert f(t) \Vert _{L^{\infty }} \leqslant C(\gamma ,s,f_{\mathrm {in}})\,. \end{aligned}$$

(1.12)

We emphasise that the constants used above do not depend on $\varvec{\varepsilon }$.

Notice that it is possible to interpolate the decay towards equilibrium in $L^1$ and estimate (1.12) in order to get a decay towards equilibrium in $L^p$, for any $p \in (1, \infty )$, for suitable initial data.

The result of no saturation described above is crucial for the LFD equation. It was obtained in [2] in the case of hard potentials using an indirect approach based on the analysis of higher regularity of solutions to ensure an $L^{\infty }$-bound independent of $\varvec{\varepsilon }$ by Sobolev embedding. In this work the approach is direct; it uses on one hand a careful study of the $L^1$ and $L^2$ moments of the solution of the equation, and on the other hand an original use of De Giorgi’s level set method, see Theorem 1.10 hereafter for more details. In both cases, a repeated use of the following technical result will be made.

Proposition 1.8

Assume that $-2< \gamma < 0$ and $f_{\mathrm{in}}$ satisfies (1.7)–(1.9) with $\varvec{\varepsilon }_{0} >0$. For any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$, any $g \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$ and any smooth and compactly supported function $\phi$, there is $C_{0} >0$ (depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$) such that

$$\begin{aligned} -\int _{{\mathbb {R}}^{3}}\phi ^{2}\varvec{c}_{\gamma }[g]\mathrm {d}v\leqslant & {} \delta \,\int _{{\mathbb {R}}^{3}}\left| \nabla \left( \langle v\rangle ^{\frac{\gamma }{2}}\phi (v)\right) \right| ^{2}\mathrm {d}v \nonumber \\&+C_{0}(1+\delta ^{\frac{\gamma }{2+\gamma }})\int _{{\mathbb {R}}^{3}}\phi ^{2}\langle v\rangle ^{\gamma }\mathrm {d}v, \qquad \forall \,\delta >0. \end{aligned}$$

(1.13)

The above inequality (1.13) has been established in [13, Theorem 2.7] with harmonic analysis tools and study of $A_{p}$-weights. This inequality is referred to as a $\delta$-Poincaré inequality in [13]. The proof of [13] can be applied without major difficulty to the Landau-Fermi-Dirac context. We nevertheless provide here an elementary proof, based in particular on Pitt’s inequality [14], with a slightly sharper estimate (1.13). On the counterpart, our method seems to apply only for the range of parameters considered here, i.e. $-2< \gamma < 0$. Related convolution inequalities will be then established in Section 2.3 and exploited for the implementation of the De Giorgi method in Section 4.

The aforementioned proposition plays a fundamental role in the establishment of the following $L^{1}$-$L^{2}$ moments estimates for the solutions to (1.11.10):

Theorem 1.9

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Assume that

$$\begin{aligned} \varvec{m}_{s}(0) < \infty , \qquad s > 4 + |\gamma |. \end{aligned}$$

Then, there exists a positive constant $\varvec{C}_{s} >0$ depending on s and $f_{\mathrm {in}}$ through $\varvec{m}_{s}(0)$, $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ such that

$$\begin{aligned} \varvec{E}_{s}(t)\leqslant \varvec{C}_{s}\left( t^{-\frac{3}{2}}+t\right) , \qquad \quad \varvec{m}_{s}(t) \leqslant \varvec{C}_{s}\left( 1+t\right) \quad t > 0\,. \end{aligned}$$

(1.14)

Moreover, there exists $\beta _{1} >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ and $\varvec{m}_{\frac{3|\gamma |}{2}}(0)$ such that, for $s > 6+|\gamma |$,

$$\begin{aligned} \varvec{C}_{s} \leqslant \beta _{1}\left[ \left( \beta _{1}s\right) ^{ {{\frac{8-\gamma }{4+2\gamma }}(s+\gamma -2)+1}}+ {2^{\frac{s}{|\gamma |}} {(1+s)^{\frac{5}{2}}} \varvec{m}_{s}(0)}\right] . \end{aligned}$$

(1.15)

It is worth noticing that Theorem 1.9 shows the instantaneous appearance of weighted $L^{2}$-norms independent of $\varvec{\varepsilon }$. Similar to hard potentials [2], we are required to investigate simultaneously the evolution of the $L^{1}$ and $L^{2}$ moments through the evolution of since the quantum parameter $\varvec{\varepsilon }$ induces a strong coupling between the two kinds of moments. Our estimate shows a linear time growth of the combined $L^{1}$ and $L^{2}$ moments which depends on the moment of order s only through the pre-factor $C_{s}$. Such a bound is fundamental for the proof of the main Theorem 1.7 which combines its slowly increasing character with an interpolation technique based upon an entropy/entropy production estimate established in [15]. The use of such an interpolation process is typical of soft potential cases for kinetic equations (and briefly described in [15]). Notice that combining the relaxation result together with the aforementioned slowly increasing bound proves, a posteriori, the uniform-in-time estimate (1.12).

In fact, to prove the no-saturation result of Theorem 1.7, the key point is the following pointwise estimate:

Theorem 1.10

Assume that $f_{\mathrm {in}}$ satisfies (1.7)–(1.9) with $\varvec{\varepsilon }_{0} >0$. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$, let f(t, v) be a weak solution to (1.11.10). Let $s > \frac{3}{2}|\gamma |$ be given and assume that $f_{\mathrm {in}} \in L^{1}_{s}({\mathbb {R}}^{3})$. Then, there is a positive constant C depending only on s, $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ such that, for any $T> t_{*} >0$,

$$\begin{aligned} \sup _{t \in [t_{*},T)}\left\| f(t)\right\| _{L^{\infty }} \leqslant C\,\Big ( 1+ t_{*}^{- \frac{3s}{4s-3|\gamma |}-\frac{3}{4}} \Big )\,\Big [\,\sup _{t \in [0,T)}\varvec{m}_{s}(t)\Big ]^{ \frac{3|\gamma |}{4s-3|\gamma |}}\,. \end{aligned}$$

(1.16)

We mentioned previously that we prove Theorem 1.10 thanks to an original use of the level set method of De Giorgi [16] which is a well-known tool for parabolic equations, see the recent surveys [17, 18], and became quite recently efficient for the study of spatially inhomogeneous kinetic Equations [13, 19]. In the spatially homogeneous situation considered here, the method has the flavour of the approach introduced in [20] for the Boltzmann equation, and recently extended to the inhomogeneous framework in [21]. The implementation of the level set method uses a new critical parameter $\gamma = - \frac{4}{3}$, which is possibly of technical nature, but could be significant even if the physical meaning of such a threshold value remains unclear. We also point out that, this special value $\gamma =-\frac{4}{3}$ is specific to the physical dimension $d=3$ and, very likely, another threshold value would appear in dimension $d \ne 3$. If $\gamma > -\frac{4}{3}$ indeed, one can pick here above $s=2$ so that $\sup _{t\geqslant 0}\varvec{m}_{s}(t) < \infty$, and, of course, (1.16) yields a pointwise estimate for f(t) independent of both T and $\varvec{\varepsilon }$, proving in a direct way the saturation property in Theorem 1.7.

It is worth noticing that a related pointwise estimate has been obtained in the classical case $\varvec{\varepsilon }=0$ in [13] for the range $-2 \leqslant \gamma <0.$ Namely, for solutions f(t) to the classical Landau equation in ${\mathbb {R}}^{3}$, [13, Theorem 2.1] asserts that there exists $C >0$ such that

$$\begin{aligned} f(t,v) \leqslant C\left( 1+\frac{1}{t}\right) ^{\frac{3}{2}}\,\langle v\rangle ^{\frac{3}{2}|\gamma |}\,\qquad \quad t > 0,\quad v \in {\mathbb {R}}^{3}. \end{aligned}$$

Clearly, our method of proof applies directly to this case and, in some sense, improves the result of [13] since combining (1.16) with the uniform bound on the moments (1.12) yields the bound

$$\begin{aligned} \sup _{t \geqslant t_{*}}\Vert f(t)\Vert _{L^{\infty }} \leqslant C\Big ( 1+{t_{*}^{- \frac{3s}{4s-3|\gamma |}-\frac{3}{4}}} \Big ), \qquad t_{*} > 0\,. \end{aligned}$$

This eliminates the need of the polynomial weight $\langle v\rangle ^{-\frac{3}{2}|\gamma |},$ at the price of a slightly worse estimate for the short-time behaviour (notice that since $s >\frac{3}{2}|\gamma |$, we have $\frac{3s}{4s-3|\gamma |}+\frac{3}{4} >\frac{3}{2}$).

We indicate that if stretched exponential moments initially exist, then the convergence towards equilibrium can be proved to have a stretched exponential rate as well, similar to related works on the Landau equation, see for example [22]. A precise result is given in Theorem 6.10. We mention here that such a result uses again interpolation technique between slowly increasing bounds for $L^{1}$ and $L^{2}$ weighted estimates for the solution to (1.11.10) and the entropy/entropy production. The slowly increasing bounds for moments associated to stretched exponential weights is deduced directly from Theorem 1.9 by exploiting the fact that we kept track of the dependence of $\varvec{C}_{s}$ in terms of s in (1.15).

1.6 Organization of the paper

After this Introduction, the paper is organized as follows. Section 2 collects several known results about the Fermi-Dirac entropy and the entropy production associated to (1.11.10) and solutions to (1.11.10). We also present in this Section the proof of the technical result stated in Proposition 1.8 as well as some other related convolution estimates. Section 3 is devoted to the study of both the $L^{1}$ and $L^{2}$ moments of solutions to (1.11.10), culminating with the proof of Theorem 1.9. In Section 4 we implement De Giorgi’s level set methods resulting in Theorem 1.10 whereas in Section 5 we collect the results of the previous sections which, combined with the study of the entropy production performed in [15], allow to derive the algebraic convergence towards equilibrium in Theorem 1.7. We upgrade this rate of convergence in Section 6 showing a stretched exponential rate of convergence for solutions associated with initial datum with finite stretched exponential moments. The paper ends with two Appendices. Appendix A is devoted to some additional regularity estimates for solutions to (1.11.10) resulting in Theorem 1.6. The full proof of Theorem 1.5 is then postponed to the Appendix B.

2 Preliminary Results

2.1 Boltzmann and Fermi-Dirac Entropy and entropy production

Recall the classical Boltzmann entropy

$$\begin{aligned} H(f)=\int _{{\mathbb {R}}^{3}}f\log f\,\mathrm {d}v\,. \end{aligned}$$

The Fermi-Dirac entropy is introduced as

$$\begin{aligned} \begin{aligned} {\mathcal {S}}_{\varvec{\varepsilon }}(f)&= - \varvec{\varepsilon }^{-1}\int _{{\mathbb {R}}^3} \Big [\varvec{\varepsilon }f\log (\varvec{\varepsilon }f)+(1-\varvec{\varepsilon }f)\log (1-\varvec{\varepsilon }f)\Big ] \, \mathrm {d}v\\&= -\varvec{\varepsilon }^{-1}\big (H(\varvec{\varepsilon }\,f)+H(1-\varvec{\varepsilon }f)\big )\,. \end{aligned} \end{aligned}$$

(2.1)

The Fermi-Dirac relative entropy is defined as follows: given nonnegative $f,\,g \in L^1_2({\mathbb {R}}^3)$ with $0 \leqslant f \leqslant \varvec{\varepsilon }^{-1}$ and $0 \leqslant g \leqslant \varvec{\varepsilon }^{-1}$, set

$$\begin{aligned} {\mathcal {H}}_{\varvec{\varepsilon }}(f|g)=-{\mathcal {S}}_{\varvec{\varepsilon }}(f)+{\mathcal {S}}_{\varvec{\varepsilon }}(g). \end{aligned}$$

For the Fermi-Dirac relative entropy, a two-sided Csiszár-Kullback inequality holds true (see [23, Theorem 3]). There exists $C >0$ (depending only on $\varvec{\varepsilon }$ and $\Vert g\Vert _{L^{1}_{2}}$) such that

$$\begin{aligned} \Vert g-{\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}}^{2} \leqslant \left( 2\int _{{\mathbb {R}}^{3}}g(v)\,\mathrm {d}v\right) \,{\mathcal {H}}_{\varvec{\varepsilon }}(g|{\mathcal {M}}_{\varvec{\varepsilon }}) \leqslant C\,\Vert g-{\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{2}}. \end{aligned}$$

(2.2)

The long time behaviour of the solutions of the equation will be studied using the classical method consisting in comparing the relative entropy with the entropy production. In our case, the entropy production is defined as

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}(g):=-\int _{{\mathbb {R}}^{3}}{\mathcal {Q}}(f)\big [\log f(v)-\log (1-\varvec{\varepsilon }f(v))\big ]\mathrm {d}v\,. \end{aligned}$$

(2.3)

One can show that

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}(g)=\frac{1}{2} \int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}\Psi (v-v_{*})\, \varvec{\** }_{\varvec{\varepsilon }}[g](v,v_{*})\,\mathrm {d}v\,\mathrm {d}v_{*}\,,\qquad \Psi (z)=|z|^{\gamma +2}\,, \end{aligned}$$

(2.4)

for any smooth function $0< g < \varvec{\varepsilon }^{-1}$, with

$$\begin{aligned} \begin{aligned} {\vec {\** }}_{\vec {\varvec{\varepsilon }}}[g](v,v_{*})&:=\Pi (v-v_{*})\Big (g_{*}(1-{\vec {\varvec{\varepsilon }}} g_{*})\nabla g - g(1-{\vec {\varvec{\varepsilon }}} g)\nabla g_{*}\Big )\\&\quad \left( \frac{\nabla g}{g(1-{\vec {\varvec{\varepsilon }}} g)}-\frac{\nabla g_{*}}{g_{*}(1-{\vec {\varvec{\varepsilon }}} g_{*})}\right) \\&=gg_{*}(1-{\vec {\varvec{\varepsilon }}} g)(1-\varvec{\varepsilon }g_{*})\left| \Pi (v-v_{*})\left( \frac{\nabla g}{g(1-{\vec {\varvec{\varepsilon }}} g)}-\frac{\nabla g_{*}}{g_{*}(1-{\vec {\varvec{\varepsilon }}} g_{*})}\right) \right| ^{2} \\&\geqslant 0\,. \end{aligned} \end{aligned}$$

(2.5)

A thorough analysis of the link between the Landau-Fermi-Dirac entropy and its entropy production ${\mathscr {D}}_{\varvec{\varepsilon }}$ has been established by the authors in a previous contribution [15], and we refer to the op. cit. for more details on the topic.

2.2 General estimates

One has the following result, refer to [2,  Lemma 2.3 & 2.4].

Lemma 2.1

Let $0\leqslant f_{\mathrm {in}}\in L^{1}_{2}({\mathbb {R}}^{3})$ be fixed and satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_{0} >0$. Then, for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$, the following hold:

• (1) For any $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$, it holds that

  $$\begin{aligned} \inf _{0<\varvec{\varepsilon }\leqslant \varvec{\varepsilon }_{0}}\int _{|v|\leqslant R(f_{\mathrm {in}})} f(1-\varvec{\varepsilon }f)\, \mathrm {d}v \geqslant \eta (f_{\mathrm {in}})>0\,, \end{aligned}$$

  (2.6)

  for some $R(f_{\mathrm {in}})>0$ and $\eta (f_{\mathrm {in}})$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm {in}})$ but not on $\varvec{\varepsilon }$.

• (2) For any $\delta >0$ there exists $\eta (\delta )>0$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm {in}})$ such that for any $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$, and any measurable set $A\subset {\mathbb {R}}^3$,

  $$\begin{aligned} |A|\leqslant \eta (\delta ) \Longrightarrow \int _A f(1-\varvec{\varepsilon }f)\, \mathrm {d}v \leqslant \delta . \end{aligned}$$

  (2.7)

A consequence of Lemma 2.1 is the following technical result which will be used for the study of moments.

Lemma 2.2

Let $0\leqslant f_{\mathrm {in}}\in L^{1}_{2}({\mathbb {R}}^{3})$ be fixed and bounded satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_{0} >0$. Let $\gamma <0$. Then, there exists $\eta _{\star } >0$ depending only on $H(f_{\mathrm {in}})$ and $\Vert f_{\mathrm{in}}\Vert _{L^1_2}$ such that, for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ and any $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$, one has

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\left( 1+|v-v_{*}|^{2}\right) ^{\frac{\gamma }{2}}\,f(v_{*}) \left( 1-\varvec{\varepsilon }\,f(v_{*})\right) \mathrm {d}v_{*}\geqslant \eta _{\star }\langle v\rangle ^{\gamma }, \qquad \forall \, v \in {\mathbb {R}}^{3}. \end{aligned}$$

(2.8)

Proof

For simplicity, given $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$, we set $F=f(1-\varvec{\varepsilon }\,f)$. From Lemma 2.1,

$$\begin{aligned} \varrho _{F}:=\int _{{\mathbb {R}}^{3}}F(v_{*})\,\mathrm {d}v_{*}\geqslant \eta (f_{\mathrm {in}}) >0. \end{aligned}$$

Let $v \in {\mathbb {R}}^{3}$ be fixed and define the probability measure $\mathrm {d}\mu$ over ${\mathbb {R}}^{3}$ by

$$\begin{aligned} \mu (\mathrm {d}v_{*})=F(v-v_{*})\frac{\mathrm {d}v_{*}}{\varrho _{F}}. \end{aligned}$$

We introduce the convex function $\Phi (r)=(1+r)^{\frac{\gamma }{2}}$, $r >0$. One has, thanks to Jensen’s inequality,

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\left( 1+|v-v_{*}|^{2}\right) ^{\frac{\gamma }{2}} F(v_{*})\mathrm {d}v_{*}= & {} \varrho _{F}\,\int _{{\mathbb {R}}^{3}}\Phi (|v_{*}|^{2})\mu (\mathrm {d}v_{*})\\\geqslant & {} \varrho _{F}\,\Phi \left( \int _{{\mathbb {R}}^{3}}|v_{*}|^{2}\mu (\mathrm {d}v_{*})\right) . \end{aligned}$$

Now,

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}|v_{*}|^{2}\mu (\mathrm {d}v_{*})=\frac{1}{\varrho _{F}}\int _{{\mathbb {R}}^{3}}|v -v_{*}|^{2}F(v_{*})\,\mathrm {d}v_{*}\leqslant 2|v|^{2}+\frac{6}{\varrho _{F}} , \end{aligned}$$

and, since $\Phi$ is nonincreasing,

$$\begin{aligned} \varrho _{F}\,\int _{{\mathbb {R}}^{3}}\Phi (|v_{*}|^{2})\mu (\mathrm {d}v_{*})\geqslant & {} \varrho _{F}\Phi \left( 2|v|^{2}+\frac{6}{\varrho _{F}}\right) \geqslant \varrho _{F}\Phi \left( \frac{6+6|v|^{2}}{\varrho _{F}}\right) \\\geqslant & {} 12^{\frac{\gamma }{2}}\varrho _{F}^{1-\frac{\gamma }{2}}\langle v\rangle ^{\gamma }\,, \end{aligned}$$

where we used that $\varrho _{F} \leqslant 1$ thanks to (1.9) and $\Phi (r)\geqslant (2r)^{\frac{\gamma }{2}}$ for $r>1$. Since $\varrho _{F}^{1-\frac{\gamma }{2}} \geqslant \eta (f_{\mathrm {in}})^{\frac{2-\gamma }{2}}$ the result follows with $\eta _{\star }= {12^{\frac{\gamma }{2}}}\eta (f_{\mathrm {in}})^{\frac{2-\gamma }{2}} >0.$ $\square$

The following coercivity estimate for the matrix $\varvec{\Sigma }[f]$ holds. Its proof is a copycat of [24, Proposition 2.3] applied to $F=f(1-\varvec{\varepsilon }f)$ after using Lemma 2.1 appropriately.

Proposition 2.3

Let $0\leqslant f_{\mathrm {in}}\in L^{1}_{2}({\mathbb {R}}^{3})$ be fixed and satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_{0} >0$. Then, there exists a constant $K_{0} > 0,$ depending on $H(f_{\mathrm {in}})$ and $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ but not $\varvec{\varepsilon }$, such that

$$\begin{aligned} \forall \, v,\, \xi \in {\mathbb {R}}^3, \qquad \sum _{i,j} \, \varvec{\Sigma }_{i,j}[f](v) \, \xi _i \, \xi _j \geqslant K_{0} \langle v \rangle ^{\gamma } \, |\xi |^2 , \end{aligned}$$

holds for any $\varvec{\varepsilon }\in [0, \varvec{\varepsilon }_0]$ and $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$.

2.3 Convolution inequalities

We establish here some of the main technical tools used in the paper. We begin with the proof of Proposition 1.8 stated in the introduction, which provides suitable estimates on the zero-th order term $\varvec{c}_{\gamma }[g]=-2(\gamma +3)|\cdot |^{\gamma }*g$.

Proof of Proposition 1.8

Let $g \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$ be fixed. For a given nonnegative $\phi$, set

$$\begin{aligned} I[\phi ] :=-\int _{{\mathbb {R}}^{3}}\phi ^{2}\varvec{c}_{\gamma }[g]\,\mathrm {d}v=2\,(\gamma +3)\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}|v-v_{*}|^{\gamma }\phi ^{2}(v)g(v_{*})\,\mathrm {d}v\,\mathrm {d}v_{*}. \end{aligned}$$

For any $v,v_{*}\in {\mathbb {R}}^{6}$, if $|v-v_{*}| < \frac{1}{2}\langle v\rangle$, then $\langle v\rangle \leqslant 2\langle v_{*}\rangle$, and we deduce from this, see [25,  Eq. (2.5)],

$$\begin{aligned} |v-v_{*}|^{\gamma }\leqslant 2^{-\gamma }\langle v\rangle ^{\gamma }\left( {\mathbf {1}}_{\left\{ |v-v_{*}| \geqslant \frac{\langle v\rangle }{2}\right\} }+ \langle v_{*}\rangle ^{-\gamma }|v-v_{*}|^{\gamma }{\mathbf {1}}_{\left\{ |v-v_{*}|< \frac{\langle v\rangle }{2}\right\} }\right) . \end{aligned}$$

(2.9)

Thanks to this inequality, we get $I[\phi ] \leqslant 6\cdot 2^{-\gamma }\left( I_{1}+I_{2}\right)$, with

$$\begin{aligned} I_{1}=\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }\phi ^{2}(v)\,\mathrm {d}v \int _{|v-v_{*}| \geqslant \frac{\langle v\rangle }{2}} g(v_{*})\,\mathrm {d}v_{*}\leqslant \, \Vert f_{\mathrm{in}}\Vert _{L^{1}} \Vert \langle \cdot \rangle ^{\gamma }\phi ^{2}\Vert _{L^{1}}\,, \end{aligned}$$

while

$$\begin{aligned} I_{2}=\int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{-\gamma }g(v_{*})\,\mathrm {d}v_{*}\int _{|v-v_{*}| < \frac{1}{2}\langle v\rangle }|v-v_{*}|^{\gamma }\langle v\rangle ^{\gamma }\phi ^{2}(v)\,\mathrm {d}v. \end{aligned}$$

Set $\psi (v)=\langle v\rangle ^{\frac{\gamma }{2}}\phi (v)$, from which we get that

$$\begin{aligned} I_{2} \leqslant \int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{-\gamma }g(v_{*})\,\mathrm {d}v_{*}\int _{{\mathbb {R}}^{3}}|v-v_{*}|^{\gamma }\psi ^{2}(v)\,\mathrm {d}v. \end{aligned}$$

According to Pitt’s inequality which reads, in ${\mathbb {R}}^{n}$, $\int _{{\mathbb {R}}^{n}}|x|^{-\alpha }|f(x)|^{2}\mathrm {d}x \lesssim \int _{{\mathbb {R}}^{n}}|\xi |^{\alpha } \,\left| {\widehat{f}}(\xi )\right| ^{2}\mathrm {d}\xi$ for any $0< \alpha < n$, [14], there is a universal constant $c >0$ such that, for any $v_{*}\in {\mathbb {R}}^{3}$,

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}|v-v_{*}|^{\gamma }\psi ^{2}(v)\,\mathrm {d}v=\int _{{\mathbb {R}}^{3}}|v|^{\gamma }|\psi (v-v_{*})|^{2}\,\mathrm {d}v \leqslant c\int _{{\mathbb {R}}^{3}}|\xi |^{-\gamma }\,\left| \widehat{\tau _{v_{*}}\psi }(\xi )\right| ^{2} \mathrm {d}\xi , \end{aligned}$$

where $\tau _{v_{*}}\psi (\cdot )=\psi (\cdot -v_{*}).$ Since $|\widehat{\tau _{v_{*}}\psi }(\xi )|=|{\widehat{\psi }}(\xi )|$, we get

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}|v-v_{*}|^{\gamma }\psi ^{2}(v)\,\mathrm {d}v \leqslant c\int _{{\mathbb {R}}^{3}}|\xi |^{-\gamma }\,|{\widehat{\psi }}(\xi )|^{2}\mathrm {d}\xi . \end{aligned}$$

This results in

$$\begin{aligned} \begin{aligned} I_{2}&\leqslant c\left( \int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{-\gamma }g(v_{*})\mathrm {d}v_{*}\right) \,\int _{{\mathbb {R}}^{3}}|\xi |^{-\gamma }\,|{\widehat{\psi }}(\xi )|^{2}\mathrm {d}\xi \\&\leqslant c\,{\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}}\,\int _{{\mathbb {R}}^{3}}|\xi |^{-\gamma }\,|{\widehat{\psi }}(\xi )|^{2}\mathrm {d}\xi =:c\,{\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}}\,J, \end{aligned} \end{aligned}$$

where we used that $-\gamma < 2$. Now, for any $R >0$, we split the above integral J in Fourier variable as

$$\begin{aligned} {J}=\int _{|\xi | < R}|\xi |^{-\gamma }|{\widehat{\psi }}(\xi )|^{2}\mathrm {d}\xi +\int _{|\xi | \geqslant R}|\xi |^{-\gamma }|{\widehat{\psi }}(\xi )|^{2}\mathrm {d}\xi ={J}_{1}+{J}_{2}. \end{aligned}$$

On the one hand, using Parseval identity, $J_{1} \leqslant R^{-\gamma }\Vert \psi \Vert _{L^{2}}^{2}=R^{-\gamma }\Vert \langle \cdot \rangle ^{\gamma }\phi ^{2}\Vert _{L^{1}}.$ On the other hand,

$$\begin{aligned} {J}_{2}=\int _{|\xi | \geqslant R}|\xi |^{-(2+\gamma )}|\xi |^{2}|{\widehat{\psi }}(\xi )|^{2}\mathrm {d}\xi \leqslant R^{-(2+\gamma )}\int _{{\mathbb {R}}^{3}}|\xi |^{2}\,|{\widehat{\psi }}(\xi )|^{2}\mathrm {d}\xi , \end{aligned}$$

that is, ${J}_{2}\leqslant R^{-(2+\gamma )}\left\| \nabla \psi \right\| _{L^{2}}^{2}.$ Thus,

$$\begin{aligned} J \leqslant R^{-\gamma }\Vert \langle \cdot \rangle ^{\gamma }\phi ^{2}\Vert _{L^{1}}+ R^{-(2+\gamma )}\Vert \nabla \psi \Vert _{L^{2}}^{2} , \end{aligned}$$

and

$$\begin{aligned} I[\phi ] \leqslant {6\cdot }2^{-\gamma }{\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}}\left( (1 +cR^{-\gamma })\Vert \langle \cdot \rangle ^{\gamma }\phi ^{2}\Vert _{L^1} +c\,R^{-(2+\gamma )}\Vert \nabla \psi \Vert _{L^{2}}^{2}\right) \end{aligned}$$

for any $R >0$. This proves (1.13) with $\delta =6\cdot 2^{-\gamma }{\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}}cR^{-(2+\gamma )}=6\cdot 2^{2-\gamma }cR^{-(2+\gamma )}.$    $\square$

An alternative version of the above estimate involving $L^{p}$-norms instead of Pitt’s inequality is given by the next Proposition, which now holds for the whole range of parameters between $(-3,0)$. In the sequel, we call the parameter $\lambda \in (-3,0)$ instead of $\gamma$ since we will apply the inequality later to $\lambda =\gamma$, $\lambda =\gamma +1$, etc..

Proposition 2.4

Let $\lambda >-3$ and $p>1$ be such that $-\lambda \,q<3$ where $\frac{1}{p}+\frac{1}{q}=1$. Then there exists $C_{p}(\lambda ) >0$ such that

$$\begin{aligned}&\left| \int _{{\mathbb {R}}^{3}}\left( |\cdot |^{\lambda } *g\right) (v)\varphi (v)\,\mathrm {d}v\right| \nonumber \\&\leqslant {\left\{ \begin{array}{ll} C_{p}(\lambda )\Vert \langle \cdot \rangle ^{-\lambda }g\Vert _{L^{1}}\Big (\Vert \langle \cdot \rangle ^{\lambda }\varphi \Vert _{L^{1}} + \Vert \langle \cdot \rangle ^{\lambda }\varphi \Vert _{L^{p}}\Big )&{}\quad \text {if }\, \lambda <0\,,\\ \\ \qquad \Vert \langle \cdot \rangle ^{\lambda }g\Vert _{L^{1}} \Vert \langle \cdot \rangle ^{\lambda }\varphi \Vert _{L^{1}}&{}\quad \text {if }\, \lambda \geqslant 0. \end{array}\right. } \end{aligned}$$

(2.10)

Proof

For $\lambda \geqslant 0$, the result is trivial since $|v-v_{*}| \leqslant \langle v\rangle \,\langle v_{*}\rangle$ for any $v,v_{*}\in {\mathbb {R}}^{3}.$ Let us consider the case $-3<\lambda < 0.$ We can assume without loss of generality that $\varphi$ and g are nonnegative. Write

$$\begin{aligned} {\mathscr {I}}:=\int _{{\mathbb {R}}^{6}}g(v_{*})\varphi (v)|v-v_{*}|^{\lambda }\mathrm {d}v\mathrm {d}v_{*}. \end{aligned}$$

Using the inequality $\langle v \rangle \leqslant \sqrt{2} \langle v - v_{*}\rangle \langle v_{*}\rangle ,$ which holds for any $v,v_{*}\in {\mathbb {R}}^{3}$, we get

$$\begin{aligned} 2^{\frac{\lambda }{2}}{\mathscr {I}}\leqslant & {} \int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{-\lambda }g(v_{*})\mathrm {d}v_{*}\int _{{\mathbb {R}}^3}|v-v_{*}|^{\lambda }\langle v - v_{*}\rangle ^{-\lambda }\langle v \rangle ^{\lambda }\varphi (v)\, \mathrm {d}v \,\\= & {} \int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{-\lambda }g(v_{*})\mathrm {d}v_{*}\int _{|v-v_{*}|<1}|v-v_{*}|^{\lambda }\langle v - v_{*}\rangle ^{-\lambda }\langle v \rangle ^{\lambda }\varphi (v)\, \mathrm {d}v \\&+\int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{-\lambda }g(v_{*})\mathrm {d}v_{*}\int _{|v-v_{*}|\geqslant 1}|v-v_{*}|^{\lambda }\langle v - v_{*}\rangle ^{-\lambda }\langle v \rangle ^{\lambda }\varphi (v)\, \mathrm {d}\\&v=:{\mathscr {I}}_{1}+{\mathscr {I}}_{2}. \end{aligned}$$

For a given $v_{*}\in {\mathbb {R}}^{3}$, on the set $\{|v-v_{*}| \geqslant 1\}$, we have $|v-v_{*}| \leqslant \langle v-v_{*}\rangle \leqslant \sqrt{2}\, |v-v_{*}|$ so that

$$\begin{aligned}&{\mathscr {I}}_{2} \leqslant 2^{-\frac{\lambda }{2}}\int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{-\lambda }g(v_{*})\mathrm {d}v_{*}\\&\int _{|v-v_{*}|\geqslant 1}\langle v \rangle ^{\lambda }\varphi (v)\, \mathrm {d}v \leqslant 2^{-\frac{\lambda }{2}}\Vert \langle \cdot \rangle ^{\lambda }\varphi \Vert _{L^{1}}\Vert \langle \cdot \rangle ^{-\lambda }g\Vert _{L^{1}}\,. \end{aligned}$$

For a given $v_{*}\in {\mathbb {R}}^{3}$, on the set $\{|v - v_{*}|<1\}$, we have $\langle v -v_{*}\rangle \leqslant \sqrt{2}$. Then, thanks to Hölder’s inequality,

$$\begin{aligned} \begin{aligned}&\int _{|v-v_{*}|< 1}|v-v_{*}|^{\lambda }\langle v - v_{*}\rangle ^{-\lambda }\langle v\rangle ^{\lambda }\varphi (v)\, \mathrm {d}v\\&\quad \leqslant 2^{-\frac{\lambda }{2}}\int _{|v-v_{*}| < 1}|v-v_{*}|^{\lambda }\langle v\rangle ^{\lambda }\varphi (v)\,\mathrm {d}v\\&\quad \leqslant 2^{-\frac{\lambda }{2}}\Vert \langle \cdot \rangle ^{\lambda }\varphi \Vert _{L^{p}}\,\left( \int _{|v-v_{*}|\leqslant 1}|v-v_{*}|^{\lambda \,q}\,\mathrm {d}v\right) ^{\frac{1}{q}}=2^{-\frac{\lambda }{2}}\left( \frac{|{\mathbb {S}}^{2}|}{3 +\lambda \,q}\right) ^{\frac{1}{q}}\Vert \langle \cdot \rangle ^{\lambda }\varphi \Vert _{L^{p}}, \end{aligned} \end{aligned}$$

from which we deduce that

$$\begin{aligned} {\mathscr {I}}_{1} \leqslant 2^{-\frac{\lambda }{2}}\left( \frac{|{\mathbb {S}}^{2}|}{3+\lambda \,q}\right) ^{\frac{1}{q}}\Vert \langle \cdot \rangle ^{\lambda }\varphi \Vert _{L^{p}}\,\Vert \langle \cdot \rangle ^{-\lambda }g\Vert _{L^{1}}. \end{aligned}$$

This gives the result with $C_{p}(\lambda ) :=2^{-\lambda }\max \left( 1,\left( \frac{|{\mathbb {S}}^{2}|}{3+\lambda \,q}\right) ^{\frac{1}{q}}\right) .$ $\square$

2.4 Consequences

An important first consequence of Proposition 1.8 is the following weighted Fisher estimate. Notice that a similar result can be deduced (for a larger range of parameters $\gamma <0$) from an alternative representation of the entropy in the spirit of [26, Theorem 2], refer to [15] for further details.

Proposition 2.5

Let $0\leqslant f_{\mathrm {in}}\in L^{1}_{2}({\mathbb {R}}^{3})$ be fixed and bounded satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_{0} >0$. Assume that $-2< \gamma <0$ and $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$. Then, there is a positive constant $C_{0}(\gamma )$ depending only on $f_{\mathrm {in}}$ through $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm {in}})$, such that for all $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$,

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\left| \nabla \sqrt{f(v)}\right| ^{2}\langle v\rangle ^{\gamma }\,\mathrm {d}v \leqslant C_{0}(\gamma )\left( 1+{\mathscr {D}}_{\varvec{\varepsilon }}(f)\right) , \qquad \forall \, \varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]. \end{aligned}$$

Proof

Let us fix $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ and $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$. Recall from (2.3) that

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}(f)=-\int _{{\mathbb {R}}^{3}}{\mathcal {Q}}(f)\left[ \log f(v)-\log (1-\varvec{\varepsilon }f(v))\right] \,\mathrm {d}v, \end{aligned}$$

where we recall that ${\mathcal {Q}}(f)=\nabla \cdot \left( \varvec{\Sigma }[f]\nabla f - \varvec{b}[f]\,F\right)$, $F=f(1-\varvec{\varepsilon }f)$. Therefore,

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}(f)= & {} \int _{{\mathbb {R}}^{3}}\left( \varvec{\Sigma }[f]\nabla f - \varvec{b}[f]\,F\right) \cdot \nabla \left[ \log f(v)-\log (1-\varvec{\varepsilon }f(v))\right] \,\mathrm {d}v\nonumber \\= & {} \int _{{\mathbb {R}}^{3}}\left( \varvec{\Sigma }[f]\nabla f - \varvec{b}[f]\,F\right) \cdot \frac{\nabla f}{F}\,\mathrm {d}v\nonumber \\= & {} \int _{{\mathbb {R}}^{3}}\frac{1}{F}\varvec{\Sigma }[f]\nabla f \cdot \nabla f\,\mathrm {d}v + \int _{{\mathbb {R}}^{3}}f \nabla \cdot \varvec{b}[f]\,\mathrm {d}v. \end{aligned}$$

(2.11)

Using Proposition 2.3, because $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$, one has

$$\begin{aligned} \varvec{\Sigma }[f]\nabla f \cdot \frac{\nabla f}{F} \geqslant \frac{K_{0}}{f}\,\langle \cdot \rangle ^{\gamma }\left| \nabla f\right| ^{2} \geqslant 4K_{0}\langle \cdot \rangle ^{\gamma }\left| \nabla \sqrt{f}\right| ^{2} , \end{aligned}$$

and, recalling $\nabla \cdot \varvec{b}[f]=\varvec{c}_{\gamma }[f]$, we deduce from (2.11) that

$$\begin{aligned} 4K_{0}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }\left| \nabla \sqrt{f(v)}\right| ^{2}\,\mathrm {d}v \leqslant {\mathscr {D}}_{\varvec{\varepsilon }}(f)-\int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f]\,f\,\mathrm {d}v. \end{aligned}$$

Then, applying Proposition 1.8, with $g=f$ and $\phi =\sqrt{f}$, there is $C_{0} >0$ such that, for any $\delta >0$,

$$\begin{aligned} 4K_{0}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }\left| \nabla \sqrt{f(v)}\right| ^{2}\,\mathrm {d}v\leqslant & {} {\mathscr {D}}_{\varvec{\varepsilon }}(f) + \delta \,\int _{{\mathbb {R}}^{3}}\left| \nabla \left( \langle v\rangle ^{\frac{\gamma }{2}}\sqrt{f(v)}\right) \right| ^{2}\,\mathrm {d}v \\&+C_{0}(1+\delta ^{\frac{\gamma }{2+\gamma }})\int _{{\mathbb {R}}^{3}}f(v)\langle v\rangle ^{\gamma }\,\mathrm {d}v. \end{aligned}$$

Using that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\left| \nabla \left( \langle v\rangle ^{\frac{\gamma }{2}}\sqrt{f(v)}\right) \right| ^{2}\,\mathrm {d}v \leqslant 2\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }\left| \nabla \sqrt{f(v)}\right| ^{2}\,\mathrm {d}v +\frac{\gamma ^{2}}{2}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }f(v)\,\mathrm {d}v , \end{aligned}$$

we can choose $\delta >0$ small enough so that

$$\begin{aligned} 2K_{0}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }\left| \nabla \sqrt{f(v)}\right| ^{2}\,\mathrm {d}v \leqslant {\mathscr {D}}_{\varvec{\varepsilon }}(f) + C_{1}(\gamma )\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }f(v)\,\mathrm {d}v \end{aligned}$$

(2.12)

for some positive constant $C_{1}(\gamma )$ depending only on $f_{\mathrm {in}}$. This gives the result. $\square$

A significant consequence of the above result is the following corollary which regards solutions to the Landau-Fermi-Dirac Equation (1.11.10).

Corollary 2.6

Assume $-2< \gamma < 0$ and let $f_{\mathrm{in}}$ be a nonnegative initial datum satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_{0} >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ and $f(t,\cdot )$ be a weak solution to Landau-Fermi-Dirac equation, then for $0< t_{1}< t_{2}$,

$$\begin{aligned} \int _{t_{1}}^{t_{2}}\mathrm {d}t \int _{{\mathbb {R}}^{3}}\left| \nabla _{v}\sqrt{f(t,v)}\right| ^{2}\langle v\rangle ^{\gamma }\,\mathrm {d}v \leqslant C_{0}(\gamma )\int _{t_{1}}^{t_{2}}\left( 1+{\mathscr {D}}_{\varvec{\varepsilon }}(f(t))\right) \mathrm {d}t, \, \end{aligned}$$

where $C_{0}(\gamma )$ is defined in Proposition 2.5. As a consequence, there exist positive constants ${\widetilde{C}}_{0}$ and ${\widetilde{C}}_{1}$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm {in}})$ so that for $0< t_{1}< t_{2}$,

$$\begin{aligned} \int _{t_{1}}^{t_{2}}\mathrm {d}t \int _{{\mathbb {R}}^{3}}\left| \nabla _{v}\left( \langle v\rangle ^{\frac{\gamma }{2}}\sqrt{f(t,v)}\right) \right| ^{2}\,\mathrm {d}v \leqslant {\widetilde{C}}_{0}(1+t_{2}-t_{1})\,, \end{aligned}$$

and

$$\begin{aligned} \int _{t_{1}}^{t_{2}}\left\| \langle \cdot \rangle ^{\gamma }f(t,\cdot )\right\| _{L^{3}}\mathrm {d}t \leqslant {\widetilde{C}}_{1}\big (1+t_{2}-t_{1}\big ) \qquad 0< t_{1} < t_{2}\,. \end{aligned}$$

(2.13)

Proof

The first inequality follows by simply integrating the inequality in Proposition 2.5. In order to get the second inequality, we use part (iii) of Definition 1.4, which ensures that

$$\begin{aligned} \int _{t_{1}}^{t_{2}}{\mathscr {D}}_{\varvec{\varepsilon }}(f(t))\mathrm {d}t \leqslant {\mathcal {S}}_{\varvec{\varepsilon }}({\mathcal {M}}_{\varvec{\varepsilon }})-{\mathcal {S}}_{\varvec{\varepsilon }}(f_{\mathrm {in}}). \end{aligned}$$

Now, on the one hand,

$$\begin{aligned} {\mathcal {S}}_{\varvec{\varepsilon }}({\mathcal {M}}_{\varvec{\varepsilon }}) = -\log \varvec{\varepsilon }-\log a_{\varvec{\varepsilon }} +3b_{\varvec{\varepsilon }} +\frac{1}{\varvec{\varepsilon }}\int _{{\mathbb {R}}^3}\log (1+\varvec{\varepsilon }M_{\varvec{\varepsilon }}(v))\,\mathrm {d}v, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{\varvec{\varepsilon }}\int _{{\mathbb {R}}^3}\log (1+\varvec{\varepsilon }M_{\varvec{\varepsilon }}(v))\,\mathrm {d}v \leqslant \int _{{\mathbb {R}}^3} M_{\varvec{\varepsilon }}(v)\,\mathrm {d}v = a_{\varvec{\varepsilon }}\left( \frac{\pi }{b_{\varvec{\varepsilon }}}\right) ^{\frac{3}{2}}. \end{aligned}$$

On the other hand,

$$\begin{aligned} {\mathcal {S}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})&= -\log (\varvec{\varepsilon }) -H(f_{\mathrm {in}})-\frac{1}{\varvec{\varepsilon }} \int _{{\mathbb {R}}^3} (1-\varvec{\varepsilon }f_{\mathrm {in}}(v)) \log (1-\varvec{\varepsilon }f_{\mathrm {in}}(v))\,\mathrm {d}v \\&\geqslant -\log \varvec{\varepsilon }-H(f_{\mathrm {in}}) . \end{aligned}$$

Hence,

$$\begin{aligned} {\mathcal {S}}_{\varvec{\varepsilon }}({\mathcal {M}}_{\varvec{\varepsilon }})-{\mathcal {S}}_{\varvec{\varepsilon }}(f_{\mathrm {in}}) \leqslant -\log a_{\varvec{\varepsilon }} +3b_{\varvec{\varepsilon }} +a_{\varvec{\varepsilon }}\left( \frac{\pi }{b_{\varvec{\varepsilon }}}\right) ^{\frac{3}{2}}+H(f_{\mathrm {in}}). \end{aligned}$$

It follows from [2,  Appendix A] that $a_{\varvec{\varepsilon }}$ and $b_{\varvec{\varepsilon }}$ are uniformly bounded with respect to $\varvec{\varepsilon }$. This means that ${\mathcal {S}}_{\varvec{\varepsilon }}({\mathcal {M}}_{\varvec{\varepsilon }})-{\mathcal {S}}_{\varvec{\varepsilon }}(f_{\mathrm {in}}) \leqslant c_{0} < \infty ,$ and the second inequality follows with ${\widetilde{C}}_{0}=C_{0}(\gamma )\max (1,c_{0})$ independent of $\varvec{\varepsilon }$. To prove (2.13), we recall that the following Sobolev inequality

$$\begin{aligned} \Vert u\Vert _{L^{6}}\leqslant C_{\mathrm {Sob}}\,\Vert \nabla u\Vert _{L^{2}}, \qquad u \in H^{1}({\mathbb {R}}^{3}), \end{aligned}$$

(2.14)

holds for some positive universal constant $C_{\mathrm {Sob}} >0$. Applying this with $u=\langle \cdot \rangle ^{\frac{\gamma }{2}}\sqrt{f(t,\cdot )}$ which is such that $\Vert u\Vert _{L^{6}}^{2}=\Vert \langle \cdot \rangle ^{\gamma }{f(t,\cdot )}\Vert _{L^{3}}$, one gets the result with ${\widetilde{C}}_{1}=C_{\mathrm {Sob}}{\widetilde{C}}_{0}.$ $\square$

One can get rid of the degenerate weight in (2.13) to get a mere $L^{p}$ bound. We refer to [13, Proposition 5.2] for a complete proof.

Lemma 2.7

Assume $-2< \gamma < 0$ and let $f_{\mathrm{in}}$ be a nonnegative initial datum satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_{0} >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ and $f(t,\cdot )$ be a weak solution to Landau-Fermi-Dirac equation. Then, there exists $C >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm {in}})$ such that for $0< t_{1} < t_{2}$,

$$\begin{aligned} \int _{t_{1}}^{t_{2}}\Vert f(t,\cdot )\Vert _{L^{p}}^{p}\mathrm {d}t \leqslant C\big (1+t_{2}-t_{1}\big ), \end{aligned}$$

holds with $p=\min \left( \frac{5}{3},\frac{3(2 +|\gamma |)}{2+3|\gamma |}\right) .$

3 Moments Estimates

We study here the evolution of both $L^{1}_{s}$ and $L^{2}_{s}$ moments of weak solutions to (1.11.10). Our goal is to prove Theorem 1.9.

3.1 $L^{1}$-Moments

We start with the following basic observation for the study of moments.

Lemma 3.1

Assume $-2< \gamma < 0$ and let $f_{\mathrm{in}}$ be a nonnegative initial datum satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_{0} >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ and $f(t,\cdot )$ be a weak solution to Landau-Fermi-Dirac equation. For any $s >2$, one has

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\int _{{\mathbb {R}}^{3}}f(t,v)\langle v\rangle ^{s}\mathrm {d}v={\mathscr {J}}_{s}(f,F)={\mathscr {J}}_{s,1}(f,F)+{\mathscr {J}}_{s,2}(f,F), \end{aligned}$$

(3.1)

where $F=f(1-\varvec{\varepsilon }f)$ and, for any nonnegative measurable map**s $h,g \geqslant 0$ and $s >2$, we use the notations

$$\begin{aligned} {\mathscr {J}}_{s,1}(h,g)&=2s\int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}h(v)g(v_{*}) \,|v-v_{*}|^{\gamma }\left( \langle v\rangle ^{s-2}-\langle v_{*}\rangle ^{s-2} \right) \\&\quad \left( |v_{*}|^{2}-(v \cdot v_{*})\right) \,\mathrm {d}v\,\mathrm {d}v_{*},\\ {\mathscr {J}}_{s,2}(h,g)&=s(s-2)\int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}\langle v \rangle ^{s-4}h(v)g(v_{*})|v-v_{*}|^{\gamma }\\&\quad \left( |v|^{2}\,|v_{*}|^{2} -(v\cdot {v_{*}})^{2}\right) \,\mathrm {d}v\,\mathrm {d}v_{*}. \end{aligned}$$

Moreover, for any nonnegative g,

$$\begin{aligned} {\mathscr {J}}_{s,1}(g,g)=2s \int \int _{{\mathbb {R}}^{3} \times {\mathbb {R}}^{3}}g(v)\,g(v_{*})\, |v-v_{*}|^{\gamma }\langle v\rangle ^{s-2}\left( \langle v_{*}\rangle ^{2}-\langle v\rangle ^{2}\right) \,\,\mathrm {d}v\,\,\mathrm {d}v_{*}. \end{aligned}$$

Proof

For a convex function $\Phi \,:\,{\mathbb {R}}^{+} \rightarrow {\mathbb {R}}^{+}$, we get from (1.11.10)

$$\begin{aligned} \dfrac{\mathrm {d}}{\mathrm {d}t}\int _{{\mathbb {R}}^{3}}f(t,v)\,\Phi (|v|^{2})\mathrm {d}v=4\int _{{\mathbb {R}}^{3}}\,\mathrm {d}v\int _{{\mathbb {R}}^{3}}f\,F_{*}|v-v_{*}|^{\gamma }\,\Lambda ^{\Phi }(v,v_{*})\mathrm {d}v_{*}, \end{aligned}$$

where $F=f(1-\varvec{\varepsilon }f)$, and

$$\begin{aligned} \Lambda ^{\Phi }(v,v_{*})= & {} \left[ |v_{*}|^{2}-(v\cdot v_{*})\right] \left[ \Phi '(|v|^{2})-\Phi '(|v_{*}|^{2})\right] \\&+\left[ |v|^{2}|v_{*}|^{2}-(v \cdot v_{*})^{2}\right] \,\Phi ''(|v|^{2}). \end{aligned}$$

Picking $\Phi (r) := (1+r)^{\frac{s}{2}}$, one sees that

$$\begin{aligned} 4\int _{{\mathbb {R}}^{3}}\,\mathrm {d}v\int _{{\mathbb {R}}^{3}}f\,F_{*}|v-v_{*}|^{\gamma }\,\Lambda ^{\Phi }(v,v_{*})\mathrm {d}v_{*}={\mathscr {J}}_{s}(f,F). \end{aligned}$$

Now, a symmetry argument shows that

$$\begin{aligned} \int \int _{{\mathbb {R}}^{3} \times {\mathbb {R}}^3}g\,g_{*}(v\cdot v_{*})\left[ \Phi '(|v|^{2})-\Phi '(|v_{*}|^{2})\right] \,|v-v_{*}|^{\gamma }\,\mathrm {d}v\,\mathrm {d}v_{*}=0 , \end{aligned}$$

that is,

$$\begin{aligned} {\mathscr {J}}_{s,1}(g,g)=2s \int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}g\,g_{*}\,|v-v_{*}|^{\gamma }\left( \langle v\rangle ^{s-2}-\langle v_{*}\rangle ^{s-2}\right) |v_{*}|^{2}\,\mathrm {d}v\,\mathrm {d}v_{*}, \end{aligned}$$

and, using symmetry again, we get

$$\begin{aligned} {\mathscr {J}}_{s,1}(g,g)=2s \int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}g\,g_{*}\,|v-v_{*}|^{\gamma }\langle v\rangle ^{s-2}\underset{=\langle v_{*}\rangle ^{2}-\langle v\rangle ^{2}}{\underbrace{\left[ |v_{*}|^{2}-|v|^{2}\right] }}\,\mathrm {d}v\\,dv_{*}, \end{aligned}$$

which gives the new expression for ${\mathscr {J}}_{s,1}(g,g)$.$\square$

Remark 3.2

According to Young’s inequality, for $s >2$ one has $\langle v\rangle ^{s-2}\langle v_{*}\rangle ^{2} \leqslant \frac{s-2}{s}\langle v\rangle ^{s}+\frac{2}{s}\langle v_{*}\rangle ^{s}$. Thus,

$$\begin{aligned} \int _{{\mathbb {R}}^{6}}g\,g_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s-2}\langle v_{*}\rangle ^{2}\,\mathrm {d}v\,\mathrm {d}v_{*}\leqslant & {} \frac{s-2}{s}\int _{{\mathbb {R}}^{6}}g\,g_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s}\,\mathrm {d}v\,\mathrm {d}v_{*}\\&+ \frac{2}{s}\int _{{\mathbb {R}}^{6}}g\,g_{*}|v-v_{*}|^{\gamma }\langle v_{*}\rangle ^{s}\,\mathrm {d}v\,\mathrm {d}v_{*}\\= & {} \int _{{\mathbb {R}}^{6}}g\,g_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s}\,\mathrm {d}v\,\mathrm {d}v_{*}, \end{aligned}$$

where we used a simple symmetry argument for the last identity. In particular, one sees that

$$\begin{aligned} {\mathscr {J}}_{s,1}(g,g) \leqslant 0. \end{aligned}$$

Let us now estimate ${\mathscr {J}}_{s}(f,F)$. The basic observation is the following:

Lemma 3.3

For any $s \geqslant 0$, $f\geqslant 0$, $F=f\,(1 - \varvec{\varepsilon }\,f)$,

$$\begin{aligned} {\mathscr {J}}_{s,1}(f,F)= & {} \frac{1}{2}\Big ({\mathscr {J}}_{s,1}(F,F) +{\mathscr {J}}_{s,1}(f,f)\Big )\nonumber \\&+\frac{\varvec{\varepsilon }}{2}\Big ({\mathscr {J}}_{s,1}(f^{2},f)-{\mathscr {J}}_{s,1}(f,f^{2})\Big ) -\frac{1}{2}\varvec{\varepsilon }^{2}{\mathscr {J}}_{s,1}(f^{2},f^{2}), \end{aligned}$$

(3.2)

with

$$\begin{aligned} {\mathscr {J}}_{s,1}(f^{2},f)-{\mathscr {J}}_{s,1}(f,f^{2})= & {} 2s\int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}f^{2}(v)f(v_{*}) |v-v_{*}|^{\gamma +2}\\&\left( \langle v\rangle ^{s-2}-\langle v_{*}\rangle ^{s-2}\right) \,\mathrm {d}v\,\mathrm {d}v_{*}. \end{aligned}$$

Proof

This is proven by direct inspection, using that $F=f(1-\varvec{\varepsilon }f)$, so that

$$\begin{aligned} {\mathscr {J}}_{s,1}(f,F)={\mathscr {J}}_{s,1}(f,f)-\varvec{\varepsilon }{\mathscr {J}}_{s,1}(f,f^{2}), \end{aligned}$$

and also

$$\begin{aligned} {\mathscr {J}}_{s,1}(f,F)= & {} {\mathscr {J}}_{s,1}(F,F)+\varvec{\varepsilon }{\mathscr {J}}_{s,1}(f^{2},F)\\= & {} {\mathscr {J}}_{s,1}(F,F)+\varvec{\varepsilon }{\mathscr {J}}_{s,1}(f^{2},f)-\varvec{\varepsilon }^{2}{\mathscr {J}}_{s,1}(f^{2},f^{2}). \end{aligned}$$

Taking the mean of these two identities gives (3.2). Now, write for simplicity

$$\begin{aligned} I :={\mathscr {J}}_{s,1}(f^{2},f)-{\mathscr {J}}_{s,1}(f,f^{2}) . \end{aligned}$$

One has

$$\begin{aligned} I= & {} 2s\int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}f(v)f(v_{*})\left( f(v)-f(v_{*})\right) \,|v-v_{*}|^{\gamma }\left( \langle v\rangle ^{s-2}-\langle v_{*}\rangle ^{s-2}\right) \\&\left( |v_{*}|^{2}-(v \cdot v_{*})\right) \,\mathrm {d}v\,\mathrm {d}v_{*}, \end{aligned}$$

from which we deduce, by a symmetry argument, that

$$\begin{aligned} I=s \int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}f(v)f(v_{*})\left( f(v) -f(v_{*})\right) |v-v_{*}|^{\gamma +2}\left( \langle v\rangle ^{s-2} -\langle v_{*}\rangle ^{s-2}\right) \,\mathrm {d}v\,\mathrm {d}v_{*}, \end{aligned}$$

which gives the desired expression using symmetry again. $\square$

We estimate separately the terms involved in (3.2) starting with the terms ${\mathscr {J}}_{s,1}(F,F)$ and ${\mathscr {J}}_{s,1}(f,f)$.

Lemma 3.4

If $f_{\mathrm {in}}$ satisfies (1.7)–(1.9) for some $\varvec{\varepsilon }_{0} >0$, for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ and any $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$, it holds

$$\begin{aligned} {\mathscr {J}}_{s,1}(F,F) \leqslant 2s\left( \Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}\int _{{\mathbb {R}}^{3}}F\langle v\rangle ^{s-2}\,\mathrm {d}v - \eta _{\star }\int _{{\mathbb {R}}^{3}}F(v)\langle v\rangle ^{s+\gamma }\,\mathrm {d}v\right) \,, \quad \forall \, s >2,\nonumber \\ \end{aligned}$$

(3.3)

where $F=f(1-\varvec{\varepsilon }\,f)$ and $\eta _{\star } >0$ is the constant in Lemma 2.2 which depends only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm {in}}).$ In the same way,

$$\begin{aligned} {\mathscr {J}}_{s,1}(f,f) \leqslant 2s\left( \Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}\int _{{\mathbb {R}}^{3}}f\langle v\rangle ^{s-2}\,\mathrm {d}v - \eta _{\star }\int _{{\mathbb {R}}^{3}}f(v)\langle v\rangle ^{s+\gamma }\,\mathrm {d}v\right) \,, \qquad \forall \, s >2.\nonumber \\ \end{aligned}$$

(3.4)

Proof

For $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$ fixed, we recall that

$$\begin{aligned} {\mathscr {J}}_{s,1}(F,F):=2s\int _{{\mathbb {R}}^{6}}F\,F_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s-2}\left( \langle v_{*}\rangle ^{2}-\langle v\rangle ^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}, \end{aligned}$$

and replace, as in [24], $|v-v_{*}|^{\gamma }$ with its regularized version $\langle v-v_{*}\rangle ^{\gamma }$. This gives

$$\begin{aligned} {\mathscr {J}}_{s,1}(F,F)= & {} 2s\int _{{\mathbb {R}}^{6}}F\,F_{*}\langle v-v_{*}\rangle ^{\gamma }\langle v\rangle ^{s-2}\left( \langle v_{*}\rangle ^{2}-\langle v\rangle ^{2}\right) \,\mathrm {d}v\,\mathrm {d}v_{*}\nonumber \\&+2s\int _{{\mathbb {R}}^{6}}F\,F_{*}\left( |v-v_{*}|^{\gamma }-\langle v-v_{*}\rangle ^{\gamma }\right) \langle v\rangle ^{s-2}\left( \langle v_{*}\rangle ^{2}-\langle v\rangle ^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}.\nonumber \\ \end{aligned}$$

(3.5)

Recall that $|v-v_{*}|^{\gamma }-\langle v-v_{*}\rangle ^{\gamma } \geqslant 0.$ Using Hölder’s inequality with the measure $\mathrm {d}\mu (v,v_{*})=F\,F_{*}\left( |v-v_{*}|^{\gamma }-\langle v-v_{*}\rangle ^{\gamma }\right) \mathrm {d}v\mathrm {d}v_{*}$ and $p=\frac{s}{s-2},$ $q=\frac{s}{2}$ so that $1/p+1/q=1$, one gets

$$\begin{aligned}&\int _{{\mathbb {R}}^{6}}F\,F_{*}\left( |v-v_{*}|^{\gamma }-\langle v-v_{*}\rangle ^{\gamma }\right) \langle v\rangle ^{s-2} \langle v_{*}\rangle ^{2}\,\mathrm {d}v\mathrm {d}v_{*}\\&\quad \leqslant \left( \int _{{\mathbb {R}}^{6}}\langle v\rangle ^{s}\mathrm {d}\mu (v,v_{*})\right) ^{\frac{s-2}{s}}\,\left( \int _{{\mathbb {R}}^{6}}\langle v_{*}\rangle ^{s}\mathrm {d}\mu (v,v_{*})\right) ^{\frac{2}{s}} , \end{aligned}$$

which, by symmetry, reads

$$\begin{aligned} \int _{{\mathbb {R}}^{6}}F\,F_{*}\left( |v-v_{*}|^{\gamma }-\langle v-v_{*}\rangle ^{\gamma }\right) \langle v\rangle ^{s-2} \langle v_{*}\rangle ^{2}\,\mathrm {d}v\mathrm {d}v_{*}\leqslant \int _{{\mathbb {R}}^{6}}\langle v\rangle ^{s}\mathrm {d}\mu (v,v_{*})\,. \end{aligned}$$

Consequently, the second term in the right-hand side of (3.5) is nonpositive. Thus,

$$\begin{aligned} {\mathscr {J}}_{s,1}(F,F)&\leqslant 2s\int _{{\mathbb {R}}^{6}}F\,F_{*}\langle v-v_{*}\rangle ^{\gamma }\langle v\rangle ^{s-2}\left( \langle v_{*}\rangle ^{2}-\langle v\rangle ^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}\\&=2s\int _{{\mathbb {R}}^{6}}F\,F_{*}\langle v-v_{*}\rangle ^{\gamma }\langle v\rangle ^{s-2}\langle v_{*}\rangle ^{2}\,\mathrm {d}v\mathrm {d}v_{*}\\&\quad -2s\int _{{\mathbb {R}}^{6}}F\,F_{*}\langle v-v_{*}\rangle ^{\gamma }\langle v\rangle ^{s}\,\mathrm {d}v\mathrm {d}v_{*}. \end{aligned}$$

For $v \in {\mathbb {R}}^{3}$ fixed, one has

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}F_{*}\langle v-v_{*}\rangle ^{\gamma }\langle v_{*}\rangle ^{2}\mathrm {d}v_{*}\leqslant \int _{{\mathbb {R}}^{3}}F_{*}\langle v_{*}\rangle ^{2}\mathrm {d}v_{*}\leqslant \Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}{=4}\,, \end{aligned}$$

whereas, thanks to Lemma 2.2, $\int _{{\mathbb {R}}^{3}}F_{*}\langle v-v_{*}\rangle ^{\gamma }\mathrm {d}v_{*}\geqslant \eta _{\star }\langle v\rangle ^{\gamma }.$ This easily gives (3.3). One proves the result in the same way for ${\mathscr {J}}_{s,1}(f,f)$, noticing that the above lower bound still holds if $f_{*}$ replaces $F_{*}$, since $f_{*}\geqslant F_{*}$.$\square$

One can evaluate the other terms in (3.2) as presented in the following lemma.

Lemma 3.5

Assume that $-2< \gamma < 0$ and let $f_{\mathrm {in}}$ satisfy (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ and $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$ be given. There is a positive constant $C_{0} >0$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^1_2}$ such that, for any $\delta \in (0,1)$ and any $s >2$,

(3.6)

whereas,

(3.7)

Proof

Recall that

$$\begin{aligned} -\varvec{\varepsilon }^{2}{\mathscr {J}}_{s,1}(f^{2},f^{2})=2s\varvec{\varepsilon }^{2}\int _{{\mathbb {R}}^{6}} f^{2}f^{2}_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s-2}\left( \langle v\rangle ^{2}-\langle v_{*}\rangle ^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}. \end{aligned}$$

Neglecting the negative term and using that $\varvec{\varepsilon }\,f^{2}_{*} \leqslant f_{*}$, we obtain

$$\begin{aligned} -\varvec{\varepsilon }^{2}{\mathscr {J}}_{s,1}(f^{2},f^{2}) \leqslant 2s\varvec{\varepsilon }\int _{{\mathbb {R}}^{6}}f^{2}(v)f_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s}\,\mathrm {d}v\mathrm {d}v_{*}. \end{aligned}$$

(3.8)

Inequality (3.6) is obtained using Proposition 1.8 with $g=f_{*}$ and $\phi (v)=\langle v\rangle ^{\frac{s}{2}}f(v)$ and noticing that $(1+\delta ^{\frac{\gamma }{2+\gamma }}) \leqslant 2\delta ^{\frac{\gamma }{2+\gamma }}$ for any $\delta \in (0,1)$ since $\frac{\gamma }{2+\gamma } <0$. The proof of (3.7) is obvious since (thanks to Lemma 3.3)

$$\begin{aligned}&\frac{\varvec{\varepsilon }}{2}\left( {\mathscr {J}}_{s,1}(f^{2},f)-{\mathscr {J}}_{s,1}(f,f^{2})\right) \\&\quad =\varvec{\varepsilon }s\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}f^{2}(v)f(v_{*}) |v-v_{*}|^{\gamma +2}\left( \langle v\rangle ^{s-2}-\langle v_{*}\rangle ^{s-2}\right) \,\mathrm {d}v\mathrm {d}v_{*}\\&\quad \leqslant \varvec{\varepsilon }s\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}f^{2}(v)f(v_{*}) |v-v_{*}|^{\gamma +2} \langle v\rangle ^{s-2}\,\mathrm {d}v\,\mathrm {d}v_{*}\\&\quad \leqslant 2\varvec{\varepsilon }s \int _{{\mathbb {R}}^{3}}f^{2}(v)\langle v\rangle ^{\gamma +s}\,\mathrm {d}v\int _{{\mathbb {R}}^{3}}f(v_{*})\langle v_{*}\rangle ^{\gamma +2}\,\mathrm {d}v_{*}, \end{aligned}$$

where we use that, since $\gamma +2 \in (0,2)$, we have $|v-v_{*}|^{\gamma +2}\leqslant 2\langle v\rangle ^{\gamma +2}\langle v_{*}\rangle ^{\gamma +2}$. This proves inequality (3.7) where the last inequality obviously comes from $\varvec{\varepsilon }f^{2} \leqslant f.$ $\square$

Let us now investigate the second term ${\mathscr {J}}_{s,2}(f,F)$ in the right-hand side of (3.1).

Lemma 3.6

Assume that $-2<\gamma < 0.$ Let $f_{\mathrm {in}}$ satisfy (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$, $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$ and $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm{in}})$, $F= f \,(1 - \varvec{\varepsilon }f)$ be given. Then for any $s >2$,

$$\begin{aligned} {\mathscr {J}}_{s,2}(f,F) \leqslant {6s(s-2)}\int _{{\mathbb {R}}^{3}}f\langle v\rangle ^{s-2}\,\mathrm {d}v\, . \end{aligned}$$

(3.9)

Proof

Let $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$ be fixed. Recall that

$$\begin{aligned} {\mathscr {J}}_{s,2}(f,F):=s(s-2)\int _{{\mathbb {R}}^{6}}f\,F_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s-4}\left( |v|^{2}\,|v_{*}|^{2}-(v\cdot v_{*})^{2}\right) \,\mathrm {d}v\,\mathrm {d}v_{*}, \end{aligned}$$

and split the integral according to $|v-v_{*}| < 1$ and $|v-v_{*}| \geqslant 1.$ Since

$$\begin{aligned} 0 \leqslant |v|^{2}\,|v_{*}|^{2}-(v\cdot v_{*})^{2} \leqslant |v|^{2}|v_{*}|^{2}\,, \end{aligned}$$

one sees that

$$\begin{aligned}&\int _{|v-v_{*}|\geqslant 1}f\,F_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s-4} \left( |v|^{2}\,|v_{*}|^{2}-(v\cdot v_{*})^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}\\&\quad \leqslant \int _{{\mathbb {R}}^{6}}f\,F_{*}\langle v\rangle ^{s-4}|v|^{2}|v_{*}|^{2}\,\mathrm {d}v\mathrm {d}v_{*}\\&\leqslant \Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}} \int _{{\mathbb {R}}^{3}}f\langle v\rangle ^{s-2}\,\mathrm {d}v{=4\int _{{\mathbb {R}}^{3}}f\langle v\rangle ^{s-2}\,\mathrm {d}v}. \end{aligned}$$

For the integral on the set $|v-v_{*}| <1$, one uses that $|v|^{2}\,|v_{*}|^{2}-(v\cdot v_{*})^{2} \leqslant |v|\,|v_{*}|\,|v-v_{*}|^{2}$ to get

$$\begin{aligned}&\int \int _{|v-v_{*}|< 1}f\,F_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s-4}\left( |v|^{2}\,|v_{*}|^{2}-(v\cdot v_{*})^{2}\right) \,\mathrm {d}v\,\mathrm {d}v_{*}\\&\quad \leqslant \int \int _{|v-v_{*}|<1}f\,F_{*}|v-v_{*}|^{\gamma +2}\langle v\rangle ^{s-4}|v|\,|v_{*}|\,\mathrm {d}v\mathrm {d}v_{*}\\&\quad \leqslant \int _{{\mathbb {R}}^{3}}f\langle v\rangle ^{s-3}\,\mathrm {d}v \int _{{\mathbb {R}}^{3}}F_{*}|v_{*}|\mathrm {d}v_{*}\,, \end{aligned}$$

where we used that $\gamma +2 \geqslant 0$ to deduce $|v-v_{*}|^{\gamma +2} \leqslant 1.$ Since, from Young’s inequality,

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}F_{*}|v_{*}|\mathrm {d}v_{*}\leqslant \frac{1}{2}\int _{{\mathbb {R}}^{3}}F_{*}\langle v_{*}\rangle ^{2}\mathrm {d}v_{*}\leqslant \frac{1}{2}\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}{=2}\,, \end{aligned}$$

one deduces the result. $\square$

We apply the previous results to solutions f(t, v) to (1.11.10) to obtain the following proposition.

Proposition 3.7

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, there is a positive constant $C >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, such that, for any $s >2$ and $\delta \in (0,1)$, there are positive constant $\varvec{K}_{s}$ which depend on s and $H(f_{\mathrm{in}})$ and $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ satisfying

(3.10)

Moreover, there exists $\beta > 0$ depending only on $H(f_{\mathrm{in}})$ and $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ such that, for $s\geqslant 3$,

$$\begin{aligned} \varvec{K}_{s} \leqslant {\beta \left( \beta (s-2)\right) ^{\frac{s-2}{\gamma +2}} \left( \frac{s-2}{s+\gamma }\right) ^{\frac{s+\gamma }{\gamma +2}} \leqslant \beta \left( \beta \,s\right) ^{\frac{s-2}{\gamma +2}}}. \end{aligned}$$

(3.11)

Proof

According to (3.1) and (3.2), one has

$$\begin{aligned} \dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{s}(t)= & {} \tfrac{1}{2}\Big ({\mathscr {J}}_{s,1}(F,F) +{\mathscr {J}}_{s,1}(f,f)\Big )\\&+\frac{\varvec{\varepsilon }}{2}\Big ({\mathscr {J}}_{s,1}(f^{2},f) -{\mathscr {J}}_{s,1}(f,f^{2})\Big ) -\frac{\varvec{\varepsilon }^{2}}{2}{\mathscr {J}}_{s,1}(f^{2},f^{2})+{\mathscr {J}}_{s,2}(f,F), \end{aligned}$$

with $f=f(t,v)$ and $F=f(1-\varvec{\varepsilon }f).$ One sees from (3.3)–(3.4) that

$$\begin{aligned} \tfrac{1}{2}\left( {\mathscr {J}}_{s,1}(F,F)+{\mathscr {J}}_{s,1}(f,f)\right)\leqslant & {} s\left( \Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}\int _{{\mathbb {R}}^{3}}(f+F)\langle v\rangle ^{s-2}\,\mathrm {d}v \right. \\&\left. -\eta _{\star }\int _{{\mathbb {R}}^{3}}(F+f)\langle v\rangle ^{s+\gamma }\,\mathrm {d}v\right) , \end{aligned}$$

whereas, from (3.6) and (3.7),

for any $\delta \in (0,1)$. Using then (3.9) to estimate ${\mathscr {J}}_{s,2}(f,F)$, we deduce that

Since $\gamma +2 >0$, the map** $v \in {\mathbb {R}}^{3} \mapsto 4\langle v\rangle ^{s-2}-\frac{1}{2}\eta _{\star }\langle v\rangle ^{s+\gamma }$ is bounded by some positive constant $\varvec{K}_{s} >0$ which depends on $f_{\mathrm {in}}$ through $\eta _{\star }$. Thus, we deduce that

with ${\bar{C}}=\max \left( \frac{C_{0}}{2},8\right) .$ Again, since $\gamma >-2$, up to a modification of $\varvec{K}_{s}$, we have $6(s-2) \varvec{m}_{s-2}(t) \leqslant \varvec{K}_{s}+\frac{\eta _{\star }}{4}\varvec{m}_{s+\gamma }(t)$, from which we easily deduce (3.10). Let us now explicit $\varvec{K}_{s}$. One observes from the aforementioned computations that one can take $\varvec{K}_{s}=\max (\sup _{x>0}u_{s}(x),\sup _{x>0}w_{s}(x))$, where

$$\begin{aligned} u_{s}(x):=4 x^{s-2}-\frac{\eta _{\star }}{2}x^{s+\gamma }\,, \qquad w_{s}(x):= 6(s-2)x^{s-2}-\frac{\eta _{\star }}{4}x^{s+\gamma }, \qquad x >0. \end{aligned}$$

It is clear that $\sup _{x >0}u_{s}(x)=u_{s}({\bar{x}})$ and $\sup _{x >0}w_{s}(x)=w_{s}({\tilde{x}})$, where

$$\begin{aligned} {\bar{x}}=\left( \frac{8(s-2)}{\eta _{\star }(s+\gamma )}\right) ^{\frac{1}{2+\gamma }}, \qquad {\tilde{x}}=\left( \frac{24(s-2)^{2}}{\eta _{\star }(s+\gamma )}\right) ^{\frac{1}{2+\gamma }}, \end{aligned}$$

and consequently, $\sup _{x>0}u_{s}(x)=4{\bar{x}}^{s-2}\frac{\gamma +2}{s+\gamma },$ $\sup _{x >0}w_{s}(x)=6(s-2){\tilde{x}}^{s-2}\frac{\gamma +2}{s+\gamma }.$ Therefore, for any $s \geqslant 3$, we see that $\varvec{K}_{s}=\sup _{x>0}w_{s}(x)$, and one checks that (3.11) holds for some explicit $\beta >0$. $\square$

3.2 $L^{2}$-Estimates

We now aim to study the evolution of weighted $L^{2}$-norms of f(t, v). Kee** previous notations, we have the lemma.

Lemma 3.8

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). For any $s \geqslant 0$, it holds that

(3.12)

where $\varvec{A}(v)=\langle v\rangle ^{2}\mathbf {Id}+(s-2)\,v\otimes v$, $v \in {\mathbb {R}}^{3}$ and $K_{0}$ is defined in Proposition 2.3.

Proof

As in [2], for any $s \geqslant 0$,

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t} \int _{{\mathbb {R}}^3} f^2(t,v) \langle v\rangle ^{s} \mathrm {d}v= & {} -\int _{{\mathbb {R}}^3} \langle v\rangle ^{s}(\varvec{\Sigma }[f] \nabla f) \cdot \nabla f \,\mathrm {d}v \\&- \, s \int _{{\mathbb {R}}^3} f \langle v\rangle ^{s-2}(\varvec{\Sigma }[f] \nabla f)\cdot v\, \mathrm {d}v \\&+ \int _{{\mathbb {R}}^3}\langle v\rangle ^{s} f(1-\varvec{\varepsilon }f) \varvec{b}[f] \cdot \nabla f \,\mathrm {d}v\\&+ \,s \int _{{\mathbb {R}}^3} (\varvec{b}[f] \cdot v )\, f^2(1-\varvec{\varepsilon }f)\langle v\rangle ^{s-2} \,\mathrm {d}v. \end{aligned}$$

Using the uniform ellipticity of the diffusion matrix $\varvec{\Sigma }[f]$ (recall Proposition 2.3), we deduce that

$$\begin{aligned} \int _{{\mathbb {R}}^3} \langle v\rangle ^{s}(\varvec{\Sigma }[f] \nabla f) \cdot \nabla f \,\mathrm {d}v \geqslant K_{0} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{s+\gamma }\,\left| \nabla f\right| ^{2}\,\mathrm {d}v\,. \end{aligned}$$

Moreover, writing

$$\begin{aligned} \nabla \left( \langle v\rangle ^{\frac{s+\gamma }{2}}\,f\right) =\langle v\rangle ^{\frac{s+\gamma }{2}}\nabla f + \frac{s+\gamma }{2}v\,\langle v\rangle ^{\frac{s+\gamma }{2}-2}f\,, \end{aligned}$$

from which

$$\begin{aligned} \langle v\rangle ^{s+\gamma }\left| \nabla f\right| ^{2} \geqslant \frac{1}{2}\left| \nabla \left( \langle v\rangle ^{\frac{s+\gamma }{2}}f\right) \right| ^{2} - (s+\gamma )^{2} \langle v\rangle ^{s+\gamma -2}f^{2}(v), \end{aligned}$$

(3.13)

we also have

$$\begin{aligned} \int _{{\mathbb {R}}^3}\langle v\rangle ^{s} f(1-\varvec{\varepsilon }f) \varvec{b}[f] \cdot \nabla f \,\mathrm {d}v= & {} - \int _{{\mathbb {R}}^3} \left( \frac{1}{2} f^2 - \frac{\varvec{\varepsilon }}{3} f^3 \right) \nabla \cdot \Big (\varvec{b}[f] \langle v\rangle ^{s}\Big ) \,\mathrm {d}v\\= & {} -s\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{s-2}\left( \frac{1}{2} f^2 - \frac{\varvec{\varepsilon }}{3} f^3 \right) \varvec{b}[f]\cdot v\,\,\mathrm {d}v \\&- \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{s}\left( \frac{1}{2} f^2 - \frac{\varvec{\varepsilon }}{3} f^3 \right) \nabla \cdot \varvec{b}[f]\, \,\mathrm {d}v. \end{aligned}$$

Therefore, recalling that $\nabla \cdot \varvec{b}[f]=\varvec{c}_{\gamma }[f]=-2(\gamma +3)|\cdot |^{\gamma }*f$, we get

Let us investigate more carefully the last term. Integration by parts shows that

$$\begin{aligned} -s\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{s-2}f \left( \varvec{\Sigma }[f]\nabla f\cdot v \right) \,\mathrm {d}v&= -\frac{s}{2}\int _{{\mathbb {R}}^{3}}\nabla f^{2} \cdot \Big (\varvec{\Sigma }[f] \, \langle v\rangle ^{s-2}v \Big ) \,\,\mathrm {d}v\\&= \frac{s}{2}\int _{{\mathbb {R}}^{3}} f^{2} \; \nabla \cdot \Big (\varvec{\Sigma }[f]\,\langle v\rangle ^{s-2} v \Big ) \,\,\mathrm {d}v\,. \end{aligned}$$

Using the product rule

$$\begin{aligned} \nabla \cdot \Big (\varvec{\Sigma }[f]\,\langle v\rangle ^{s-2} v \Big )=\langle v\rangle ^{s-2}\,{\varvec{B}}[f]\cdot v\, + \mathrm {Trace}\left( \varvec{\Sigma }[f] \cdot \, D_{v}\left( \langle v\rangle ^{s-2} v\right) \right) , \end{aligned}$$

where $D_{v}\big ( \langle v\rangle ^{s-2} v\big )$ is the matrix with entries $\partial _{v_{i}}\big ( \langle v\rangle ^{s-2} v_{j}\big )$, $i,j=1,2,3$, or more compactly,

$$\begin{aligned} D_{v}\big ( \langle v\rangle ^{s-2} v\big )=\langle v\rangle ^{s-4}\varvec{A}(v)\,, \end{aligned}$$

one gets the desired inequality, recalling that $\varvec{B}[f]=\varvec{b}[f]-\varvec{\varepsilon }\,\varvec{b}[f^{2}].$ $\square$

We deduce from the previous arguments

Proposition 3.9

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). There exists some positive constant ${\bar{C}}(f_{\mathrm {in}})$ depending on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$, such that

(3.14)

holds for any $s \geqslant 0.$

Proof

We denote by $I_{1},I_{2},I_{3},I_{4},I_{5}$ the various terms on the right-hand-side of (3.12), i.e.

and we control each term starting from $I_{1}$. Since $0 \leqslant \frac{1}{6}f^{2} \leqslant \frac{1}{2}f^{2}-\frac{\varvec{\varepsilon }}{3}f^{3} \leqslant \frac{1}{2}f^{2}$, one has

$$\begin{aligned} |I_{1}|\leqslant (\gamma +3) \int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}|v-v_{*}|^{\gamma }f^{2}(t,v)\langle v\rangle ^{s}f(t,v_{*})\mathrm {d}v_{*}\,\mathrm {d}v, \end{aligned}$$

so that, using Proposition 1.8 with $g=f(t)$ and $\phi ^{2}=\langle \cdot \rangle ^{s}f^{2}(t)$, we deduce that, for any $\delta \in (0,1)$,

where $C_{1}$ depends on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$. For the term $I_{2}$, since $0\leqslant \frac{1}{3}f^{2} \leqslant f^{2}-\frac{2\varvec{\varepsilon }}{3}f^{3} \leqslant f^{2}$, it holds that

$$\begin{aligned} |I_{2}|\leqslant & {} s\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{s-1}f^{2}(t,v)\,|\varvec{b}[f(t)](v)|\,\mathrm {d}v \\\leqslant & {} 2s\int _{{\mathbb {R}}^{6}}\langle v\rangle ^{s-1}f^{2}(t,v)|v-v_{*}|^{\gamma +1}f(t,v_{*})\mathrm {d}v_{*}\,\mathrm {d}v\,. \end{aligned}$$

Therefore, if $\gamma +1 <0$, applying Proposition 1.8 with $\varvec{c}_{\gamma +1}[g]$ instead of $\varvec{c}_{\gamma }[g]$, and $g=f(t,v)$, $\phi ^{2}=\langle \cdot \rangle ^{s-1}f^{2}(t)$, we get

whereas, if $\gamma +1 >0$, one has obviously . In both cases, for any $\delta >0$,

In the same way,

$$\begin{aligned}&|I_{4}| \leqslant \frac{\varvec{\varepsilon }s}{2}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{s-1}f^{2}(t,v)\,|\varvec{b}[f^{2}(t, \cdot )](v)|\,\mathrm {d}v\\&\leqslant \frac{s}{2}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{s-1}f^{2}(t,v)\,|\varvec{b}[f(t)](v)|\,\mathrm {d}v, \end{aligned}$$

since $\varvec{\varepsilon }f^{2}\leqslant f.$ Then, as before, for any $\delta >0$, there is $C_{1} >0$ such that

For the term $I_{5}$, one checks easily that, for any $i,j \in \{1,2,3\}$,

$$\begin{aligned} \left| \varvec{\Sigma }_{i,j}[f]\right| \leqslant 2|\cdot |^{\gamma +2}*f, \qquad \left| \varvec{A}_{i,j}(v)\right| \leqslant s\langle v\rangle ^{2}, \end{aligned}$$

and

$$\begin{aligned} |I_{5}| \leqslant 9s^{2}\int _{{\mathbb {R}}^{6}}\langle v\rangle ^{s-2}f^{2}(t,v)|v-v_{*}|^{\gamma +2}f(t,v_{*})\,\mathrm {d}v\mathrm {d}v_{*}. \end{aligned}$$

One has, since $\gamma +2 >0$,

Finally, it is easy to see that . Overall, recalling mass and energy conservation to estimate all the weighted $L^{1}$-terms, one sees that, for any ${\delta \in (0,1)}$, there is some positive constant $C(f_{\mathrm {in}})$ depending on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ (through $K_{0}$) such that

(3.15)

For $s \in [0,1]$, (3.15) can be rephrased simply as

and, picking $\delta \in (0,1)$ such that $3\delta \leqslant \frac{K_{0}}{4}$, one deduces that

(3.16)

for some positive constant ${\widetilde{C}}_{\gamma }(f_{\mathrm{in}})$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ and $\gamma$. For $s >1$, since $2s+1 \leqslant 3s$, choosing $\delta :=\min \left( \frac{K_{0}}{16s},1\right)$ we deduce from (3.15) that there is $C_{\gamma }(f_{\mathrm {in}})$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ and $\gamma >0$ such that

(3.17)

From Nash inequality, there is some universal constant $C >0$ such that

which, thanks to Young’s inequality, implies that there is $C >0$ such that, for any $\alpha >0$,

(3.18)

Choosing now $\alpha >0$ such that ${\widetilde{C}}_{\gamma }(f_{\mathrm{in}})\alpha =\frac{K_{0}}{8}$ if $s \in [0,1]$ or $C_{\gamma }(f_{\mathrm {in}})\,\left( s^{2}+{s^{-\frac{\gamma }{2+\gamma }}}{+s+s^{\frac{2}{\gamma +3}}}\right) \alpha = \frac{K_{0}}{8}$ whenever $s >1$, we end up with

where, according to estimate (3.17) and (3.18),

$$\begin{aligned} {C_{s}(f_{\mathrm{in}})=C\alpha ^{-\frac{3}{2}}C_{\gamma }(f_{\mathrm {in}})\, \left( s^{2}+{s^{-\frac{\gamma }{2+\gamma }}}{+s+s^{\frac{2}{\gamma +3}}}\right) },\qquad {s >1} \end{aligned}$$

and the last choice of $\alpha =\frac{K_{0}}{8C_{\gamma }(f_{\mathrm{in}})}\left( s^{2}+{s^{-\frac{\gamma }{2+\gamma }}}{+s+s^{\frac{2}{\gamma +3}}}\right) ^{-1}$ gives that

$$\begin{aligned} C_{s}(f_{\mathrm {in}})=C(f_{\mathrm {in}})\left( s^{2}+{s^{-\frac{\gamma }{2+\gamma }}} {+s+s^{\frac{2}{\gamma +3}}}\right) ^{{\frac{5}{2}}} \leqslant {\bar{C}}(f_{\mathrm {in}}){\left( 1+s^{\frac{10}{2+\gamma }}\right) }, \end{aligned}$$

since $\max (1,2,\frac{-\gamma }{2+\gamma },\frac{2}{3+\gamma }) \leqslant \frac{4}{2+\gamma }$ for any $-2< \gamma <0$ and with $C(f_{\mathrm {in}})$ and ${\bar{C}}(f_{\mathrm {in}})$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm {in}})$ but not on s. This shows (3.14). $\square$

Corollary 3.10

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Given $s \in [0,4+|\gamma |]$ there exists some positive constant $C(f_{\mathrm {in}})$ depending on $f_{\mathrm {in}}$ only through $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$, and such that

holds for any $0 \leqslant t_{1} < t_{2}$.

Proof

When $\frac{\gamma +s}{2}\leqslant 2$, it holds that $\Vert \langle \cdot \rangle ^{\frac{\gamma +s}{2}}f\Vert _{L^1}\leqslant \Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$, which gives the statement after integration of (3.14).$\square$

3.3 Short-time estimates and appearance of $L^{2}$-moments

Before trying to get global-in-time estimates for both $\varvec{m}_{s}(t)$ and (with a growing rate independent of s), let us start with the following non-optimal growth that has to be interpreted here as a short-time estimate.

Proposition 3.11

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, for any $s \geqslant 3$,

$$\begin{aligned} \varvec{m}_{s}(t) \leqslant \Big [ \varvec{m}_{s}(0)^{\frac{|\gamma |}{s}} + C(f_{\mathrm{in}})\,|\gamma |\,s \, t\Big ]^{\frac{s}{|\gamma |}}\,,\qquad t\geqslant 0\,, \end{aligned}$$

(3.19)

where the constant $C(f_{\mathrm {in}})$ depends on $f_{\mathrm {in}}$ only through $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ but does not depend on s. If $s \in (2,3)$ (3.19) still holds for $\gamma \in [-1,0)$ whereas, for $\gamma \in (-2,-1)$,

$$\begin{aligned} \varvec{m}_{s}(t) \leqslant \varvec{m}_{s}(0) + C(f_{\mathrm{in}}) \, t\,,\qquad t\geqslant 0\,, \end{aligned}$$

(3.20)

for a constant $C(f_{\mathrm{in}})$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^1_2}$.

Proof

Recall that, according to (3.1), $\dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{s}(t)={\mathscr {J}}_{s,1}(f,F)+{\mathscr {J}}_{s,2}(f,F)$ where, for any $s \geqslant 2$,

$$\begin{aligned} {\mathscr {J}}_{s,1}(f,F)= & {} \frac{1}{2}\Big ({\mathscr {J}}_{s,1}(F,F) +{\mathscr {J}}_{s,1}(f,f)\Big )\\&+\frac{\varvec{\varepsilon }}{2}\Big ({\mathscr {J}}_{s,1}(f^{2},f)-{\mathscr {J}}_{s,1}(f,f^{2})\Big ) -\frac{\varvec{\varepsilon }^{2}}{2}{\mathscr {J}}_{s,1}(f^{2},f^{2})\,. \end{aligned}$$

Recall (see Remark 3.2) that

$$\begin{aligned} {\mathscr {J}}_{s,1}(F,F)\leqslant 0,\,\,\qquad {\mathscr {J}}_{s,1}(f,f)\leqslant 0\,. \end{aligned}$$

(3.21)

We neglect such absorption terms since we are interested here in the short time propagation of moments. We also recall that, according to (3.7),

$$\begin{aligned} {\mathscr {J}}_{s,1}(f^{2},f)-{\mathscr {J}}_{s,1}(f,f^{2}) \leqslant \frac{{4}s}{\varvec{\varepsilon }}\,\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}\,\varvec{m}_{s+\gamma }(t)\,. \end{aligned}$$

(3.22)

Now, to deal with the term ${\mathscr {J}}_{s,1}(f^{2},f^{2})$, we observe that there is $c >0$ (independent of s) such that

$$\begin{aligned} \Big | \langle v\rangle ^{s-2}-\langle v_{*}\rangle ^{s-2} \Big | \leqslant c(s-2)\Big ( \langle v\rangle ^{s-3} + \langle v_{*}\rangle ^{s-3}\Big )\big | v - v_{*}\big |\,, \end{aligned}$$

(3.23)

which implies,

$$\begin{aligned} \begin{aligned}&\Big (\langle v\rangle ^{s-2} - \langle v_{*}\rangle ^{s-2}\Big ) \left( \langle v\rangle ^{2}-\langle v_{*}\rangle ^{2}\right) \\&\quad \leqslant c(s-2) |v-v_{*}|^2 \Big ( \langle v\rangle ^{s-3} + \langle v_{*}\rangle ^{s-3}\Big ) (\langle v\rangle + \langle v_{*}\rangle ) \\&\quad \leqslant 3c(s-2) |v-v_{*}|^2 (\langle v\rangle ^{s-2} + \langle v_{*}\rangle ^{s-2}) \qquad \text{ for } s\geqslant 3. \end{aligned} \end{aligned}$$

where we used that $a^{s-3}b \leqslant \frac{s-3}{s-2}a^{s-2}+\frac{1}{s-2}b^{s-2} \leqslant a^{s-2}+b^{s-2}$ for any $a,b >0$, $s \geqslant 3$ in the last estimate. Using then a symmetry argument, this yields

$$\begin{aligned} \begin{aligned} -{\mathscr {J}}_{s,1}(f^{2},f^{2})&=2s\int _{{\mathbb {R}}^{6}}f^{2}f^{2}_{*}|v-v_{*}|^{\gamma }\langle v\rangle ^{s-2}\left( \langle v\rangle ^{2}-\langle v_{*}\rangle ^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}\\&=s\int _{{\mathbb {R}}^{6}}f^{2}f^{2}_{*}|v-v_{*}|^{\gamma }\Big (\langle v\rangle ^{s-2} - \langle v_{*}\rangle ^{s-2}\Big ) \left( \langle v\rangle ^{2}-\langle v_{*}\rangle ^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}\\&\leqslant 3c\,\frac{s(s-2)}{\varvec{\varepsilon }^2}\int _{{\mathbb {R}}^{6}}f_{*}\,f\left( \langle v\rangle ^{s-2}+\langle v_{*}\rangle ^{s-2}\right) |v-v_{*}|^{\gamma +2}\mathrm {d}v\mathrm {d}v_{*}\\&\leqslant 6c\,\frac{s(s-2)}{\varvec{\varepsilon }^2}\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}\,\varvec{m}_{s+\gamma }(t)\,. \end{aligned} \end{aligned}$$

(3.24)

Therefore, adding estimate (3.21), (3.22), and (3.24), there is some $C_{1} >0$ such that

$$\begin{aligned} {\mathscr {J}}_{s,1}(f,F)\leqslant {C_{1}}\,s(s-1) \,\varvec{m}_{s+\gamma }(t)\,,\qquad \forall \, s\geqslant 3\,. \end{aligned}$$

(3.25)

Furthermore, recall from Lemma 3.6 that

$$\begin{aligned} {\mathscr {J}}_{s,2}(f,F) \leqslant {C_{2}}\, s(s-2) \varvec{m}_{s-2}(t). \end{aligned}$$

Consequently, there exists $C(f_{\mathrm{in}}) >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ such that

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{s}(t) \leqslant C(f_{\mathrm{in}})\,s(s-1)\varvec{m}_{s+\gamma }(t) \leqslant C(f_{\mathrm{in}})\,s(s-1)\big (\varvec{m}_{s}(t)\big )^{\frac{s+\gamma }{s}}\,, \end{aligned}$$

for any $t \geqslant 0$ and any $s \geqslant 3.$ This leads to (3.19) after integration. Let us now investigate the case $s \in (2,3)$. If $\gamma \in [-1,0)$, one simply uses that

$$\begin{aligned} \Big (\langle v\rangle ^{s-2} - \langle v_{*}\rangle ^{s-2}\Big ) \left( \langle v\rangle ^{2}-\langle v_{*}\rangle ^{2}\right) \leqslant |v-v_{*}| \Big ( \langle v\rangle ^{s-2} + \langle v_{*}\rangle ^{s-2}\Big ) (\langle v\rangle + \langle v_{*}\rangle ) \end{aligned}$$

to obtain

$$\begin{aligned} \begin{aligned} -{\mathscr {J}}_{s,1}(f^{2},f^{2})&\leqslant \,\frac{3s}{\varvec{\varepsilon }^2}\int _{{\mathbb {R}}^{6}}f_{*}\,f\left( \langle v\rangle ^{s-1}+\langle v_{*}\rangle ^{s-1}\right) |v-v_{*}|^{\gamma +1}\mathrm {d}v\mathrm {d}v_{*}\\&\leqslant \frac{6s}{\varvec{\varepsilon }^2}\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}\,\varvec{m}_{s+\gamma }(t)\,. \end{aligned} \end{aligned}$$

This estimate is similar to (3.24) and yields again (3.19). For $s\in (2,3)$ and $\gamma \in (-2,-1)$, (3.23) implies

$$\begin{aligned} \Big (\langle v\rangle ^{s-2} - \langle v_{*}\rangle ^{s-2}\Big ) \left( \langle v\rangle ^{2}-\langle v_{*}\rangle ^{2}\right) \leqslant 2 c(s-2) |v-v_{*}|^2 (\langle v\rangle + \langle v_{*}\rangle ) , \end{aligned}$$

which yields

$$\begin{aligned} \begin{aligned} -{\mathscr {J}}_{s,1}(f^{2},f^{2})&\leqslant 2c\,\frac{s(s-2)}{\varvec{\varepsilon }^2}\int _{{\mathbb {R}}^{6}}f_{*}\,f\left( \langle v\rangle +\langle v_{*}\rangle \right) |v-v_{*}|^{\gamma +2}\,\mathrm {d}v\mathrm {d}v_{*}\\&\leqslant 4c\,\frac{s(s-2)}{\varvec{\varepsilon }^2}\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}\,\varvec{m}_{3+\gamma }(t)\leqslant \frac{12c}{\varvec{\varepsilon }^2}\Vert f_{\mathrm{in}}\Vert _{L^1_2}\,\varvec{m}_{3+\gamma }(t)\,. \end{aligned} \end{aligned}$$

(3.26)

We have $\varvec{m}_{s+\gamma }(t)\leqslant \varvec{m}_{2}(t)$ and $\varvec{m}_{3+\gamma }(t)\leqslant \varvec{m}_{2}(t)$ since $s \in (2,3)$ and $\gamma \in (-2,-1)$. Consequently, adding estimate (3.21), (3.22) and (3.26) leads to

$$\begin{aligned} {\mathscr {J}}_{s,1}(f,F)\leqslant {C_{1}} \,\varvec{m}_{2}(t)\,, \end{aligned}$$

for some $C_1 >0$ depending on $\Vert f_{\mathrm{in}}\Vert _{L^1_2}$ (recall $s \in (2,3)$). Then,

$$\begin{aligned} \begin{aligned} \dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{s}(t)&\leqslant C_1\,\varvec{m}_{2}(t)+{\mathscr {J}}_{s,2}(f,F) \leqslant C_1\,\varvec{m}_2(t) + {C_{2}}\, s(s-2) \varvec{m}_{s-2}(t)\\&\leqslant C_1\varvec{m}_2(0) + 3C_2\varvec{m}_{1}(t) \leqslant \left( C_1+3C_2\right) \varvec{m}_{2}(0)=:C(f_{\mathrm{in}})\,. \end{aligned} \end{aligned}$$

This yields the desired estimate after integration for $s \in (2,3)$ and $\gamma \in (-2,-1)$. $\square$

Notice that, besides the above Corollary 3.10, one can also provide the following appearance and short-time bounds for :

Proposition 3.12

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Assume additionally that

$$\begin{aligned} \varvec{m}_{\frac{2s-3\gamma }{4}}(0) < \infty , \qquad {s > 4+|\gamma |}. \end{aligned}$$

Then, there exists a constant $c_s(f_{\mathrm {in}})$ such that

(3.27)

with moreover,

$$\begin{aligned} c_{s}(f_{\mathrm{in}}) \leqslant C(f_{\mathrm {in}})2^{\frac{s}{|\gamma |}} \left[ \varvec{m}_{\frac{2s-3\gamma }{4}}^{2}(0)+\varvec{C}_{0}^{\frac{s}{|\gamma |}} \right] \,(1 + s^{\frac{6}{2+\gamma }}) \qquad \forall {s > 6+|\gamma |,} \end{aligned}$$

(3.28)

for some positive constants $C(f_{\mathrm{in}}), \varvec{C}_{0}$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ (but not on s).

Proof

Let us pick $s \geqslant 0$ and set $T_{s}:=\frac{1}{1+s}$. Recall estimate (3.14)

for some positive constant $C(f_{\mathrm {in}})$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$. Using a classical interpolation inequality (see (4.6) in the next section), one has

where we used Sobolev’s inequality (2.14). Thus,

For $s> 4-\gamma >4+\frac{3\gamma }{2}$, we estimate $\varvec{m}_{\frac{2s-3\gamma }{4}}(t)$ and $\varvec{m}_{\frac{s+\gamma }{2}}(t)$ using Proposition 3.11. We assume for simplicity that both $\frac{2s-3\gamma }{4}$ and $\frac{s+\gamma }{2}$ are larger than 3 to use (3.19) only. This amounts to pick $s > 6+|\gamma |$. One has

$$\begin{aligned} \begin{aligned} \varvec{m}_{\frac{2s-3\gamma }{4}}(t)&\leqslant 2^{\frac{s}{2|\gamma |}}\Big (\varvec{m}_{\frac{2s-3\gamma }{4}}(0)+\big (C(f_{\mathrm{in}})|\gamma |\tfrac{2s-3\gamma }{4}t\big )^{\frac{s}{2|\gamma |}+\frac{3}{4}}\Big )\\&\leqslant 2^{\frac{s}{2|\gamma |}}\Big (\varvec{m}_{\frac{2s-3\gamma }{4}}(0)+\big (C(f_{\mathrm{in}})|\gamma |\big )^{\frac{s}{2|\gamma |}+\frac{3}{4}}\Big )\,,\qquad t \in \left( 0,T_{s}\right] , \end{aligned} \end{aligned}$$

(3.29)

and, in the same way, for $s>4-\gamma$,

$$\begin{aligned} \varvec{m}^2_{\frac{s+\gamma }{2}}(t) \leqslant \,2^{\frac{s}{|\gamma |}-2}\Big ( \varvec{m}^{2}_{\frac{s+\gamma }{2}}(0)+\big (C(f_{\mathrm{in}})|\gamma |\big )^{\frac{s}{|\gamma |}-1}\Big )\,, \qquad t \in \left( 0,T_{s}\right] \, \end{aligned}$$

(note that $\varvec{m}_{\frac{s+\gamma }{2}}(0)<\infty$ because $\frac{s+\gamma }{2}<\frac{2s-3\gamma }{4}$). Therefore,

(3.30)

where

$$\begin{aligned} \begin{aligned} \frac{1}{\varvec{a}_{s}(f_{\mathrm{in}})}&=2^{\frac{2s}{3|\gamma |}{+ {\frac{7}{3}}}}\,C^2_{\mathrm {Sob}}\,K^{-1}_0\,\Big (\varvec{m}^{\frac{4}{3}}_{\frac{2s -3\gamma }{4}}(0)+\big (C(f_{\mathrm{in}})|\gamma |\big )^{\frac{2s}{3|\gamma |}+1}\Big )\,,\\ \varvec{k}_{s}(f_{\mathrm{in}})&=C(f_{\mathrm {in}})\left( 1+s^{\frac{10}{2+\gamma }}\right) \,2^{\frac{s}{|\gamma |} {-1} }\Big ( \varvec{m}^{2}_{\frac{s+\gamma }{2}}(0)+\big (C(f_{\mathrm{in}})|\gamma |\big )^{\frac{s}{|\gamma |}-1}\Big )\,. \end{aligned} \end{aligned}$$

(3.31)

The conclusion then follows by a comparison argument. Namely, introducing

$$\begin{aligned} \varvec{\zeta }(x)={\varvec{k}_{s}}(f_{\mathrm {in}})-\varvec{a}_{s}(f_{\mathrm {in}})x^{\frac{5}{3}}, \qquad x >0\,, \end{aligned}$$

and ${\bar{x}}=\left( \frac{{2\varvec{k}_{s}}(f_{\mathrm {in}})}{\varvec{a}_{s}(f_{\mathrm {in}})}\right) ^{\frac{3}{5}}$, one has $\varvec{\zeta }(x) \leqslant -\frac{\varvec{a}_{s}(f_{\mathrm{in}})}{2}x^{\frac{5}{3}}$ for $x \geqslant {\bar{x}}$. Fixing $t_{\star }\in \left( 0,T_{s}\right]$, one has according to (3.30) that

Three cases may occur:

• i) if , then since $\varvec{\zeta }({\bar{x}}) <0$, one has for any $t \geqslant t_{\star },$

• ii) if and for any $t \in (t_{\star },T_{s})$, then one has

  

  which, after integration, yields

  

• iii) if and for some ${\overline{t}}\in (t_{\star },T_{s})$ then, setting

  

  one has, as in the second point, that

  

  Since by continuity, we deduce that for all $t \geqslant T_{\star }$ from the first point.

In all the cases it holds that

from which the result follows by letting $t_{\star } \rightarrow 0$ and with

$$\begin{aligned} c_{s}(f_{\mathrm {in}})= \max \left\{ \left( \frac{3}{\varvec{a}_{s}(f_{\mathrm{in}})}\right) ^{\frac{3}{2}}\,,\,{2^{\frac{3}{5}}} \left( \frac{\varvec{k}_{s}(f_{\mathrm{in}})}{\varvec{a}_{s}(f_{\mathrm{in}})}\right) ^{\frac{3}{5}} \right\} \end{aligned}$$

with $\varvec{a}_{s}(f_{\mathrm {in}})$ and $\varvec{k}_{s}(f_{\mathrm {in}})$ defined in (3.31) with constant $C(f_{\mathrm{in}})$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$. In particular, as far as the dependence with respect to s is concerned, we easily derive (3.28).

If $\min \left( \frac{2s-3\gamma }{4},\frac{s+\gamma }{2}\right) < 3$, then one has to estimate $\varvec{m}_{\frac{2s-3\gamma }{4}}(t)$ and/or $\varvec{m}_{\frac{s+\gamma }{2}}(t)$ using (3.20). The same computations as before allows then to end up again with (3.30) (with slightly different expression for $\varvec{k}_{s}(f_{\mathrm{in}})$ and $\varvec{a}_{s}(f_{\mathrm{in}})$) and get the result as in the previous case. Details are left to the reader. $\square$

3.4 Combined estimates

We now introduce

Combining Proposition 3.9 with Proposition 3.7, one gets

Lemma 3.13

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, for any $s >2$, there are positive constants $\varvec{K}_{s}, C_{1,s}$ which depend on s and $f_{\mathrm {in}}$ (through $H(f_{\mathrm{in}})$ and $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$) such that

(3.32)

where ${\bar{C}}(f_{\mathrm{in}})$ is the constant in inequality (3.14), $\varvec{K}_{s}$ was estimated in (3.11), and

$$\begin{aligned} C_{1,s}={\bar{C}}_{1}\big (s^{\frac{2}{2+\gamma }}+s\big ), \end{aligned}$$

for some positive constant ${\bar{C}}_{1}$ depending only on $f_{\mathrm {in}}$ through $K_{0}$ and $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$.

Proof

We simply apply (3.10) with $\delta =\min \left( {\frac{K_{0}}{16s}},1\right)$ and add the obtained inequality with (3.14) to obtain the result. We derive easily the estimate for $C_{1,s}$ since, for s large enough, $s\delta =\frac{K_{0}}{16}.$$\square$

We have all in hands to prove Theorem 1.9 in the introduction.

Proof of Theorem 1.9

Let $s > 4+|\gamma |$. Since $\frac{2s-3\gamma }{4} \leqslant s$, one has $\max \left( \varvec{m}_{s}(0),\varvec{m}_{\frac{2s-3\gamma }{4}}(0)\right) =\varvec{m}_{s}(0) < \infty$, and one deduces from Propositions 3.11 and 3.12 that

$$\begin{aligned} \varvec{E}_{s}(t) \leqslant {\bar{C}}_{s}t^{-\frac{3}{2}}, \qquad t \in \left( 0,\tfrac{1}{1+s}\right] \,, \end{aligned}$$

with

$$\begin{aligned} {\bar{C}}_{s}=\left[ \varvec{m}_{s}(0)^{\frac{|\gamma |}{s}}+C(f_{\mathrm{in}})|\gamma |\right] ^{\frac{s}{|\gamma |}} + {\frac{1}{2}c_{s}(f_{\mathrm{in}})}, \end{aligned}$$

where $c_{s}(f_{\mathrm{in}})$ and $C(f_{\mathrm{in}})$ are defined in Propositions 3.12 and  3.11. Since $s> 4 +|\gamma |$, we use (3.19) and not (3.20). In particular, using that

$$\begin{aligned} \varvec{m}_{\frac{2s-3\gamma }{4}}^{2}(0) \leqslant \varvec{m}_{\frac{3|\gamma |}{2}}(0)\varvec{m}_{s}(0) \end{aligned}$$

thanks to Cauchy-Schwarz inequality, we deduce from (3.28) that, for $s>6+|\gamma |$,

$$\begin{aligned} {\bar{C}}_{s}\leqslant & {} 2^{\frac{s+\gamma }{|\gamma |}}\left( \varvec{m}_{s}(0)+\left( {C(f_{\mathrm{in}})}|\gamma |\right) ^{\frac{s}{|\gamma |}}\right) + {C(f_{\mathrm{in}})2^{\frac{s}{|\gamma |}}\varvec{m}_{\frac{3|\gamma |}{2}}(0)\varvec{m}_{s}(0)} \,{(1 + s^{\frac{6}{2+\gamma }})} \\&+{C(f_{\mathrm{in}})\left( {2} \varvec{C}_{0}\right) ^{\frac{s}{|\gamma |}}}\, {(1 + s^{\frac{6}{2+\gamma }})}. \end{aligned}$$

Consequently, there are positive constants $C_{0},C_{1} >0$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that

$$\begin{aligned} {\bar{C}}_{s} \leqslant C_{0}2^{\frac{s}{|\gamma |}}\varvec{m}_{\frac{3|\gamma |}{2}}(0)\varvec{m}_{s}(0) + C_{1}^{\frac{s}{|\gamma |}}, \qquad s >6+|\gamma |, \end{aligned}$$

(3.33)

where we used that $1\leqslant \varvec{m}_{\frac{3|\gamma |}{2}}(0)$. Let us then focus on the evolution of $\varvec{E}_{s}(t)$ for $t \geqslant \tfrac{1}{1+s}.$ We start with (3.32) and estimate and $\varvec{m}_{\frac{s+\gamma }{2}}^{2}(t)$ as in the proof of Proposition 3.9 (see also [2,  Lemma 3.5]). Namely, as seen at the end of the proof of Prop. 3.9, there is a universal constant $C >0$ independent of s such that, for any $\delta >0$,

Inserting this in (3.32) and choosing $\delta >0$ such that $C_{1,s}\delta ={\frac{K_{0}}{32}}$, one has

where

$$\begin{aligned} {\bar{C}}_{3,s}=C\,\left( {\frac{K_{0}}{32}}\right) ^{-\frac{3}{2}} C_{1,s}^{\frac{5}{2}}{+{\bar{C}}(f_{\mathrm {in}})\left( 1+s^{\frac{10}{2+\gamma }}\right) }. \end{aligned}$$

(3.34)

Now, for $s \geqslant 4 -\gamma$,

$$\begin{aligned} \varvec{m}_{\frac{s+\gamma }{2}}(t)^{2}\leqslant \varvec{m}_{2}(t)^{\frac{s+\gamma }{s+\gamma -2}}\,\varvec{m}_{s+\gamma }(t)^{\frac{s+\gamma -4}{s+\gamma -2}} \leqslant \Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}^{\frac{s+\gamma }{s+\gamma -2}}\varvec{m}_{s+\gamma } (t)^{\frac{s+\gamma -4}{s+\gamma -2}}\,,\qquad t \geqslant 0\,,\nonumber \\ \end{aligned}$$

(3.35)

where we used the conservation of mass and energy. From Young’s inequality, for any $\delta _{0} >0$, one has then

$$\begin{aligned} \varvec{m}_{\frac{s+\gamma }{2}}(t)^{2} \leqslant \Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}^{\frac{s+\gamma }{2}}\,\delta _{0}^{-\frac{s+\gamma -4}{2}} + \delta _{0}\,\varvec{m}_{s+\gamma }(t)\,,\qquad t >0\,. \end{aligned}$$

Choosing now $\delta _{0} >0$ so that ${\bar{C}}_{3,s}\delta _{0}=s\frac{\eta _{\star }}{8}$, we end up with

(3.36)

where

$$\begin{aligned} \overline{\varvec{C}}_{s}= {2s}\varvec{K}_{s} + {{\bar{C}}_{3,s}}\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}^{\frac{s+\gamma }{2}}\, \delta _{0}^{-\frac{s+\gamma -4}{2}}= {2s}\varvec{K}_{s}+ {{\bar{C}}_{3,s}}\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}^{\frac{s+\gamma }{2}} \left( \frac{8}{\eta _{\star }}\frac{{\bar{C}}_{3,s}}{s}\right) ^{\frac{s+\gamma -4}{2}}. \end{aligned}$$

(3.37)

Integrating this inequality on $\left( \tfrac{1}{1+s},t\right)$ gives

$$\begin{aligned} \varvec{E}_{s}(t) \leqslant \varvec{E}_{s}\left( \tfrac{1}{1+s}\right) +\overline{\varvec{C}}_{s}\left( t-\tfrac{1}{1+s}\right) , \qquad t\geqslant \tfrac{1}{1+s}\,, \end{aligned}$$

so that

$$\begin{aligned} \varvec{E}_{s}(t) \leqslant {\bar{C}}_{s}\left( \tfrac{1}{1+s}\right) ^{-\frac{3}{2}}+\overline{\varvec{C}}_{s}t \leqslant {\bar{C}}_{s}(1+s)^{\frac{5}{2}}t+\overline{\varvec{C}}_{s}t, \qquad t\geqslant \tfrac{1}{1+s}\,. \end{aligned}$$

Proposition  3.11 gives now the second part of (1.14) for small times whereas Proposition 3.12 and (3.36) lead to the second part of (1.14) for large times with

$$\begin{aligned} \varvec{C}_{s} :=\max \left( {\bar{C}}_{s},{\bar{C}}_{s}(1+s)^{\frac{5}{2}} +\overline{\varvec{C}}_{s}\right) ={\bar{C}}_{s}(1+s)^{\frac{5}{2}}+\overline{\varvec{C}}_{s}. \end{aligned}$$

Combining (3.34) with Lemma 3.13, one sees first that there is $C_{3} >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that,

$$\begin{aligned} {\bar{C}}_{3,s} \leqslant C_{3} {\left( s^{\frac{5}{2}}+s^{\frac{5}{2+\gamma }}+s^{\frac{10}{2+\gamma }}\right) \leqslant 2C_{3}s^{\frac{10}{2+\gamma }}}, \qquad s >2. \end{aligned}$$

Then, using (3.11) and (3.37), one deduces that there exists $\beta _{0} >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that

$$\begin{aligned} \overline{\varvec{C}}_{s} \leqslant \beta _{0}\left[ \left( \beta _{0}\,s\right) ^{ {\frac{s-2}{\gamma +2}}+1}+\left( \beta _{0}\,s\right) ^{{\frac{8-\gamma }{4+2\gamma }(s+\gamma -2)+1}}\right] , \qquad s \geqslant 4-\gamma \end{aligned}$$

where we roughly estimate $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}^{\frac{s+\gamma }{2}}$ as $\beta _{0}^{\frac{8-\gamma }{4+2\gamma }(s+\gamma -2)+1}$ once we notice that $\frac{s+\gamma }{2} \leqslant {\frac{8-\gamma }{4+2\gamma }(s+\gamma -2)+1}$ for $s \geqslant 4-\gamma$. Combining these estimates with (3.33) and because $\frac{s-2}{\gamma +2} < \frac{8-\gamma }{4+2\gamma }(s+\gamma -2)$ for $s > 4-\gamma$, one deduces easily the estimate (1.15). $\square$

Remark 3.14

Of course, if $f_{\mathrm {in}}$ is actually belonging to $L^{1}_{s}({\mathbb {R}}^{3}) \cap L^{2}_{s}({\mathbb {R}}^{3})$, then $\varvec{E}_{s}(0) <\infty$ and one deduces after integration of (3.36) that

$$\begin{aligned} \varvec{E}_{s}(t) \leqslant \varvec{E}_{s}(0)+ \overline{\varvec{C}}_{s} \, t, \qquad t \geqslant 0. \end{aligned}$$

The above result shows the linear growth of both the $L^{1}$-moments and $L^{2}$-moments. Actually, it is possible to sharpen the above estimates (for small s) with the following proposition.

Proposition 3.15

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, for any $s \in [0,\frac{8+3\gamma }{2}]$,

• (1) if $f_{\mathrm {in}} \in L^{2}_{s}({\mathbb {R}}^{3})$, there is a positive constant $\varvec{C}_{\mathrm {in}}$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ such that

  $$\begin{aligned} \sup _{t\geqslant 0}\Vert f(t)\Vert _{L^{2}_{s}}=\sup _{t\geqslant 0}\left\| \langle \cdot \rangle ^{\frac{s}{2}}f(t,\cdot )\right\| _{L^{2}} \leqslant \varvec{C}_{\mathrm {in}}. \end{aligned}$$

  (3.38)

• (2) There are constants $\varvec{C}_{0,\mathrm {in}}$ and $\tilde{\varvec{C}}_{\mathrm {in}}$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, and $H(f_{\mathrm{in}})$ such that for any $t>0$,

  

  (3.39)

Proof

Let us pick $s \in [0,\frac{8+3\gamma }{2}]$. In light of estimate (3.14), since $\frac{s+\gamma }{2} \leqslant 2,$ we have that

for some positive constant $\varvec{K}(f_{\mathrm {in}})$ depending only on $f_{\mathrm {in}}$ through $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$. Arguing exactly as in the proof of Proposition 3.12, Equation (3.30) but with $\varvec{m}_{\frac{2s-3\gamma }{4}}\leqslant \Vert f_{\mathrm {in}}\Vert _{L^1_2}$, one deduces that

(3.40)

where we set $\kappa _{\mathrm {in}}=K_{0}\left( {4\,C_{\mathrm {Sob}}^{2}\Vert f_{\mathrm{in}} \Vert ^{\frac{4}{3}}_{ L^{1}_{2}}}\right) ^{-1}$. The first point follows then by a simple comparison argument choosing , whereas the second point is obtained exactly as in the proof of Proposition 3.12. $\square$

The following corollary is a simple consequence of Proposition 3.15:

Corollary 3.16

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, there exists a positive constant $C(f_{\mathrm {in}})$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that

$$\begin{aligned} \left| \varvec{b}[f(t)](v)\right| \leqslant C(f_{\mathrm {in}})\,\langle v\rangle ^{\max (0,1+\gamma )}{\left( 1+t^{-\frac{3}{2}}\right) , \qquad \forall \,v \in {\mathbb {R}}^{3},\quad t > 0.} \end{aligned}$$

(3.41)

Proof

Recall that

$$\begin{aligned} \left| \varvec{b}[f(t)](v)\right|= & {} 2\left| \int _{{\mathbb {R}}^{3}}(v -v_{*})|v-v_{*}|^{\gamma }f(t,v_{*})\mathrm {d}v_{*}\right| \\\leqslant & {} 2\int _{{\mathbb {R}}^{3}}|v-v_{*}|^{1+\gamma }f(t,v_{*})\mathrm {d}v_{*}\,. \end{aligned}$$

If $1+\gamma \geqslant 0$, one notices that $|v-v_{*}|^{1+\gamma } \leqslant 2^{\frac{1+\gamma }{2}}\langle v\rangle ^{1+\gamma }\langle v_{*}\rangle ^{1+\gamma }$, so that

$$\begin{aligned} \left| \varvec{b}[f(t)](v)\right| \leqslant 2^{\frac{3+\gamma }{2}}\langle v\rangle ^{1+\gamma }\varvec{m}_{1+\gamma }(t) \leqslant 2^{\frac{3+\gamma }{2}}\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}\langle v\rangle ^{1+\gamma }, \end{aligned}$$

(3.42)

since $1+\gamma <2.$ Let us now investigate the case $1+\gamma < 0.$ One splits the integral defining $\varvec{b}[f(t)](v)$ according to $|v-v_{*}| \leqslant 1$ and $|v-v_{*}|>1$. One has then, thanks to Cauchy-Schwarz inequality,

$$\begin{aligned} \begin{aligned} |\varvec{b}[f(t)](v)|&\leqslant 2\int _{{\mathbb {R}}^{3}}f(t,v_{*})\mathrm {d}v_{*}+ 2\int _{|v-v_{*}|<1}|v-v_{*}|^{\gamma +1}f(t,v_{*})\mathrm {d}v_{*}\\&\leqslant 2\Vert f(t)\Vert _{L^{1}}+2\Vert f(t)\Vert _{L^{2}}\,\left( \int _{|v-v_{*}|<1}|v-v_{*}|^{2\gamma +2}\mathrm {d}v_{*}\right) ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

Noticing that

$$\begin{aligned} \int _{|v-v_{*}|<1}|v-v_{*}|^{2\gamma +2}\mathrm {d}v_{*}=|{\mathbb {S}}^{2}|\int _{0}^{1}r^{2(\gamma +2)}\mathrm {d}r < \infty , \qquad 2+\gamma >0, \end{aligned}$$

is independent of v, one gets

$$\begin{aligned} \left| \varvec{b}[f(t)](v)\right| \leqslant C\,\big (\Vert f(t)\Vert _{L^{1}}+\Vert f(t)\Vert _{L^{2}}\big ) \end{aligned}$$

(3.43)

for some universal positive constant $C >0.$ We deduce then (3.41) from the conservation of mass and Proposition 3.15. $\square$

Estimate (3.41) implies of course that ${\sup _{t\geqslant 1}}\left| \varvec{b}[f(t)] \cdot v\right| \leqslant C(f_{\mathrm {in}})\,\langle v\rangle ^{\max (1,2+\gamma )}$. Additionally, we need the following estimate:

Lemma 3.17

Assume $-2< \gamma <0$. There exist two constants $c_{0},c_{1} >0$ depending only on $\gamma$ such that, for any nonnegative $g \in L^{1}_{2+\gamma }({\mathbb {R}}^{3})$

$$\begin{aligned} \big |\varvec{b}[g]\cdot v\big | \leqslant c_{0}\langle v\rangle ^{\gamma +2}\Vert g\Vert _{L^{1}_{2+\gamma }}-c_{1}\,\langle v\rangle ^{2}\varvec{c}_{\gamma }[g](v)\,, \end{aligned}$$

(3.44)

where we recall that $-\varvec{c}_{\gamma }[g](v)=2(\gamma +3)\int _{{\mathbb {R}}^{3}}|v-v_{*}|^{\gamma }g(v_{*})\mathrm {d}v_{*}\geqslant 0.$

Proof

Let $\delta >0$ be given. One writes

$$\begin{aligned} \begin{aligned} \varvec{b}[g] \cdot v&=-2\int _{{\mathbb {R}}^{3}}v \cdot (v-v_{*})\,|v-v_{*}|^{\gamma }g(v_{*})\mathrm {d}v_{*}\\&=-\int _{{\mathbb {R}}^{3}}|v-v_{*}|^{\gamma }g(v_{*})\left( |v-v_{*}|^{2}+|v|^{2}-|v_{*}|^{2}\right) \mathrm {d}v_{*}. \end{aligned} \end{aligned}$$

Since $\left| \,|v|^{2}-|v_{*}|^{2}\right| \leqslant \frac{1}{2}|v-v_{*}|^{2}+\frac{1}{2}|v+v_{*}|^{2} \leqslant \frac{3}{2}|v-v_{*}|^{2}+4|v|^{2}$, we get

$$\begin{aligned} |\varvec{b}[g]\cdot v| \leqslant \frac{5}{2}\int _{{\mathbb {R}}^{3}}|v-v_{*}|^{\gamma +2}g(v_{*})\mathrm {d}v_{*}+4\langle v\rangle ^{2}\int _{{\mathbb {R}}^{3}}|v-v_{*}|^{\gamma }g(v_{*})\mathrm {d}v_{*}, \end{aligned}$$

which gives the result using that

$$\begin{aligned} |v-v_{*}|^{\gamma +2}\leqslant 2^{\frac{\gamma +2}{2}}\langle v\rangle ^{\gamma +2}\langle v_{*}\rangle ^{\gamma +2}\,,\quad \text {for}\; -2< \gamma <0\,, \end{aligned}$$

and recalling the definition of $\varvec{c}_{\gamma }[g]$. $\square$

4 De Giorgi’s Approach to Pointwise Bounds

We introduce, as in [20], for any fixed $\ell \in (0,\frac{1}{2\varvec{\varepsilon }})$,

$$\begin{aligned} f_{\ell }(t,v) :=(f(t,v)-\ell ), \qquad f_{\ell }^{+}(t,v) :=f_{\ell }(t,v){\mathbf {1}}_{\{f\geqslant \ell \}}. \end{aligned}$$

To prove an $L^{\infty }$ bound for f(t, v), one looks for an $L^{2}$-bound for $f_{\ell }$. We start with the following estimate.

Lemma 4.1

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). There exist $c_{0},C_{0} >0$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^1_2}$ and $H(f_{\mathrm{in}})$ such that, for any $\ell \in (0,\frac{1}{2\varvec{\varepsilon }})$,

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert f_{\ell }^{+}(t)\Vert _{L^{2}}^{2} + c_{0}\int _{{\mathbb {R}}^{3}}\left| \nabla \left( \langle v \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(t,v)\right) \right| ^{2}\,\mathrm {d}v \\&\quad \leqslant C_{0}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(t)\Vert _{L^{2}}^{2}-\ell \int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f](t,v)\,f_{\ell }^{+}(t,v)\mathrm {d}v.\end{aligned}\end{aligned}$$

(4.1)

Proof

Given $\ell \in (0,\frac{1}{2\varvec{\varepsilon }})$, one has $\partial _{t}\left( f_{\ell }^{+}\right) ^{2}=2f_{\ell }^{+}\partial _{t}f_{\ell }^{+}=2f_{\ell }^{+}\partial _{t}f$ and $\nabla f_{\ell }^{+}={\mathbf {1}}_{\{f\geqslant \ell \}}\nabla f$, so that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert f_{\ell }^{+}(t)\Vert _{L^2}^{2}= & {} -\int _{{\mathbb {R}}^{3}}\varvec{\Sigma }\nabla f\cdot \nabla f_{\ell }^{+} \,\mathrm {d}v +\int _{{\mathbb {R}}^{3}}f(1-\varvec{\varepsilon }f)\varvec{b}[f]\cdot \nabla f_{\ell }^{+}\,\mathrm {d}v\\= & {} -\int _{{\mathbb {R}}^{3}}\varvec{\Sigma }\nabla f^{+}_{\ell }\cdot \nabla f_{\ell }^{+}\,\mathrm {d}v +\int _{{\mathbb {R}}^{3}}f(1-\varvec{\varepsilon }f)\varvec{b}[f]\cdot \nabla f_{\ell }^{+}\,\mathrm {d}v. \end{aligned}$$

Now, one easily checks that

$$\begin{aligned} \begin{aligned} f(1-\varvec{\varepsilon }f)\nabla f^{+}_{\ell }&={\mathbf {1}}_{\{f\geqslant \ell \}}\left( f_{\ell }(1-{\vec {\varvec{\varepsilon }}} f)\nabla f_{\ell }^{+} + \ell (1-{\vec {\varvec{\varepsilon }}} f)\nabla f_{\ell }^{+}\right) \\&=(1-2 \varvec{\varepsilon }\ell )f^{+}_{\ell } \nabla f^{+}_{\ell } -{\vec {\varvec{\varepsilon }}} (f^{+}_{\ell })^{2} \nabla f^{+}_{\ell } +\ell (1-{\vec {\varvec{\varepsilon }}} \ell )\nabla f^{+}_{\ell }\\&=(\tfrac{1}{2}-{\vec {\varvec{\varepsilon }}} \ell )\nabla (f^{+}_{\ell })^{2}-\frac{\varvec{\varepsilon }}{3}\nabla (f^{+}_{\ell })^{3} + \ell (1-{\vec {\varvec{\varepsilon }}}\ell )\nabla f^{+}_{\ell }. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert f_{\ell }^{+}(t)\Vert _{L^2}^{2}+\int _{{\mathbb {R}}^{3}}\varvec{\Sigma }\nabla f^{+}_{\ell }\cdot \nabla f^{+}_{\ell }\,\mathrm {d}v\\&\quad = (\tfrac{1}{2}-{\vec {\varvec{\varepsilon }}}\ell )\int _{{\mathbb {R}}^{3}}\varvec{b}[f] \cdot \nabla (f^{+}_{\ell })^{2} \,\mathrm {d}v \\&\qquad + \ell (1-{\vec {\varvec{\varepsilon }}}\ell )\int _{{\mathbb {R}}^{3}}\varvec{b}[f]\cdot \nabla f^{+}_{\ell }\,\mathrm {d}v -\frac{\varvec{\varepsilon }}{3}\int _{{\mathbb {R}}^{3}}\varvec{b}[f]\cdot \nabla (f^{+}_{\ell })^{3}\,\mathrm {d}v\\&\quad = -(\tfrac{1}{2}-{\vec {\varvec{\varepsilon }}} \ell )\int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f](f^{+}_{\ell })^{2}\,\mathrm {d}v - \ell (1-{\vec {\varvec{\varepsilon }}} \ell )\int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f] f^{+}_{\ell }\mathrm {d}v\\&+\frac{\varvec{\varepsilon }}{3}\int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f]\,(f^{+}_{\ell })^{3}\,\mathrm {d}v. \end{aligned}$$

Using that $\tfrac{1}{2}-\varvec{\varepsilon }\ell >0$ and $-\varvec{c}_{\gamma }[f] \geqslant 0$, we deduce from Proposition 2.3 and Proposition 1.8 with $\phi =f_{\ell }^{+}$ and $\delta >0$ small enough that there exist positive constants $c_{0}, C_{0}$ depending on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert f_{\ell }^{+}(t)\Vert ^{2}_{L^2} + c_{0}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }\left| \nabla f^{+}_{\ell }(t,v)\right| ^{2}\,\mathrm {d}v\\&\leqslant C_{0}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f^{+}_{\ell }(t)\Vert _{L^2}^{2} -\ell \int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f]\,f_{\ell }^{+}\,\mathrm {d}v. \end{aligned}$$

Notice that, using again inequality (3.13), we can replace easily the above with (4.1) with a different (but irrelevant) choice of $c_{0},C_{0}$. $\square$

Inspired by De Giorgi’s iteration method introduced for elliptic Equations [16], the crucial point in the level set approach of [20] is to compare some suitable energy functional associated to $f_{\ell }^{+}$ with the same energy functional at some different level $f_{k}^{+}.$ The key observation here is that, if $0 \leqslant k <\ell$, then

$$\begin{aligned} 0 \leqslant f^{+}_{\ell }\leqslant f^{+}_{k},\quad \text {and}\quad {\mathbf {1}}_{\{f_{\ell } \geqslant 0\}} \leqslant \frac{f^{+}_{k}}{\ell - k}\,. \end{aligned}$$

Indeed, on the set $\{f_{\ell }\geqslant 0\}$, one has $f \geqslant \ell >k$, so that $f^{+}_{k}=f-k=f^{+}_{\ell }+(\ell -k)$ and $\frac{f^{+}_{k}}{\ell -k}=\frac{f^{+}_{\ell }}{\ell -k}+1 \geqslant 1.$ In particular, for any $\alpha \geqslant 0$, we deduce that

$$\begin{aligned} {\mathbf {1}}_{\{f_{\ell }\geqslant 0\}}=\left( {\mathbf {1}}_{\{f_{\ell }\geqslant 0\}}\right) ^{\alpha } \leqslant \left( \frac{f^{+}_{k}}{\ell -k}\right) ^{\alpha }, \end{aligned}$$

which, since $f_{\ell }^{+}\leqslant f^{+}_{k}$, gives

$$\begin{aligned} f_{\ell }^{+} \leqslant \left( \ell -k\right) ^{-\alpha }\,\left( f^{+}_{k}\right) ^{1+\alpha } \qquad \forall \,\alpha \geqslant 0, \qquad 0 \leqslant k < \ell .\end{aligned}$$

(4.2)

On this basis, we need the following interpolation inequality where the dependence of $f_{\ell }^{+}$ with respect to time is omitted hereafter:

Lemma 4.2

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). There exists $C >0$ (independent of $\varvec{\varepsilon }$ and $f_{\mathrm{in}}$) such that, for any $0 \leqslant k < \ell$, one has

$$\begin{aligned} \Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}\Vert _{L^{2}}^{2} \leqslant C\,(\ell -k)^{-\frac{4}{3}}\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{k}^{+}\right) \right\| _{L^{2}}^{2}\,\left\| f^{+}_{k}\right\| _{L^{2}}^{\frac{4}{3}}.\nonumber \\ \end{aligned}$$

(4.3)

Moreover, for $p \in [1,3)$, there is $C_{p} >0$ such that,

$$\begin{aligned} \Vert \langle \cdot \rangle ^{\gamma }f_{\ell }^{+}\Vert _{L^{p}} \leqslant C_{p}(\ell -k)^{-(\frac{2}{p}+\frac{1}{3})}\,\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{k}^{+}\right) \right\| _{L^{2}}^{2}\,\Vert f_{k}^{+}\Vert _{L^{2}}^{\frac{2}{p}-\frac{2}{3}}, \qquad 0 \leqslant k < \ell . \end{aligned}$$

(4.4)

Finally, for any $q \in \left( \frac{8}{3},\frac{10}{3}\right)$, there is $c_{q} >0$ such that

$$\begin{aligned} \Vert f_{\ell }^{+}\Vert _{L^{2}}^{2} \leqslant \frac{c_{q}}{(\ell -k)^{q-2}}\,\left\| \langle \cdot \rangle ^{s}f_{k}^{+}\right\| _{L^{1}}^{\frac{10}{3}-q}\,\Vert f^{+}_{k}\Vert _{L^{2}}^{2(q-\frac{8}{3})}\,\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}\,f^{+}_{k}\right) \right\| _{L^{2}}^{2}, \quad \quad 0 \leqslant k < \ell \,, \end{aligned}$$

(4.5)

with $s=-\frac{3\gamma }{10-3q} > -\frac{3}{2}\gamma$.

Proof

The proof is based on the interpolation inequality

$$\begin{aligned} \Vert \langle \cdot \rangle ^{a}g\Vert _{L^{r}} \leqslant \Vert \langle \cdot \rangle ^{a_{1}}g\Vert _{L^{r_{1}}}^{\theta }\,\Vert \langle \cdot \rangle ^{a_{2}}g\Vert _{L^{r_{2}}}^{1-\theta }\,, \end{aligned}$$

(4.6)

with

$$\begin{aligned} \frac{1}{r}=\frac{\theta }{r_{1}}+\frac{1-\theta }{r_{2}}, \quad a=\theta \,a_{1}+(1-\theta )a_{2}, \quad \theta \in (0,1). \end{aligned}$$

Moreover, for the special case $r_{1}=6$, $r_{2}=2$, $a_{1}=\frac{\gamma }{2}$ and $r \in (2,6)$, thanks to Sobolev embedding, the identity will become

$$\begin{aligned} \begin{aligned} \Vert \langle \cdot \rangle ^{a}g\Vert _{L^{r}}&\leqslant C_{\theta }\,\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}g\right) \right\| _{L^{2}}^{\theta }\,\Vert \langle \cdot \rangle ^{a_{2}}g\Vert _{L^{2}}^{1-\theta },\\ \frac{1}{r}&=\frac{3-2\theta }{6}, \qquad a=\theta \,\frac{\gamma }{2}+(1-\theta )a_{2}, \qquad \theta \in (0,1),\qquad r \in (2,6). \end{aligned} \end{aligned}$$

(4.7)

With these tools in had, one has for $0 \leqslant k < \ell$ and $r >2$, writing $r=2+2\alpha$ with (4.2),

$$\begin{aligned} \Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}\Vert _{L^{2}}^{2}= & {} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }(f^{+}_{\ell }(t,v))^{2}\,\mathrm {d}v\\\leqslant & {} (\ell -k)^{-2\alpha }\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }(f^{+}_{k}(t,v))^{2+2\alpha }\,\mathrm {d}v\\&=(\ell -k)^{-(r-2)}\left\| \langle \cdot \rangle ^{\frac{\gamma }{r}}f^{+}_{k}(t)\right\| _{L^{r}}^{r}\,, \end{aligned}$$

so that (4.7) gives, with $a=\frac{\gamma }{r}$,

$$\begin{aligned} \Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}\Vert _{L^{2}}^{2} \leqslant C(\ell -k)^{-(r-2)}\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{k}^{+}(t)\right) \right\| _{L^{2}}^{r\theta }\,\left\| \langle \cdot \rangle ^{a_{2}} f^{+}_{k}\right\| _{L^{2}}^{r(1-\theta )}, \end{aligned}$$

with $\theta =\frac{3r-6}{2r}$ and $a_{2}=\frac{\gamma }{2}\frac{10-3r}{6-r}$. One picks then $r=\frac{10}{3}$ so that $a_{2}=0$ and $r\theta =2$, to obtain (4.3). One proceeds in the same way to estimate $\Vert \langle \cdot \rangle ^{\gamma }f_{\ell }^{+}\Vert _{L^{p}}^{p}$. Namely, for $r >p$,

$$\begin{aligned} \Vert \langle \cdot \rangle ^{\gamma }f_{\ell }^{+}\Vert _{L^{p}}^{p} \leqslant (\ell -k)^{-(r-p)}\left\| \langle \cdot \rangle ^{\frac{\gamma \,p}{r}}f^{+}_{k}\right\| _{L^{r}}^{r} \end{aligned}$$

and, with $r >2p$, imposing in (4.7) $a_{2}=0$ and $a=\frac{\gamma \,p}{r}$, we get $\theta =\frac{2p}{r}$ and

$$\begin{aligned} \Vert \langle \cdot \rangle ^{\gamma }f_{\ell }^{+}\Vert _{L^{p}}^{p} \leqslant C(\ell -k)^{-(r-p)}\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{k}^{+}\right) \right\| _{L^{2}}^{2p}\, \left\| f_{k}^{+}\right\| _{L^{2}}^{r-2p}, \end{aligned}$$

which gives (4.4) when $r=2+\frac{4p}{3}$ (notice that $r >2p$ since $p <3$).

Let us now prove (4.5). Let us consider first $q >2$ and use (4.6). One has

$$\begin{aligned} \Vert g\Vert _{L^{q}}\leqslant \Vert \langle \cdot \rangle ^{s}\,g\Vert _{L^{1}}^{\theta _{1}}\,\Vert g\Vert _{L^{2}}^{\theta _{2}}\,\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}g\Vert _{L^{6}}^{\theta _{3}}, \end{aligned}$$

with $\theta _{i} \geqslant 0$ $(i=1,2,3)$ such that

$$\begin{aligned} \theta _{1}+\theta _{2}+\theta _{3}=1, \qquad s\,\theta _{1}+0\cdot \theta _{2}+\frac{\gamma }{2}\theta _{3}=0, \qquad \frac{\theta _{1}}{1}+\frac{\theta _{2}}{2}+\frac{\theta _{3}}{6}=\frac{1}{q}. \end{aligned}$$

Imposing $q\theta _{3}=2$, this easily yields

$$\begin{aligned} q\theta _{1}=\frac{10}{3}-q, \qquad q\theta _{2}=2\left( q-\frac{8}{3}\right) , \qquad s=-\frac{3\gamma }{10-3q}, \qquad q \in \left( \frac{8}{3},\frac{10}{3}\right) . \end{aligned}$$

Using Sobolev inequality, this gives, for any $q \in \left( \frac{8}{3},\frac{10}{3}\right)$, the existence of a positive constant $C >0$ such that

$$\begin{aligned} \Vert g\Vert _{L^{q}}^{q} \leqslant C\,\Vert \langle \cdot \rangle ^{s}g\Vert _{L^{1}}^{\frac{10}{3}-q}\,\Vert g\Vert _{L^{2}}^{2(q-\frac{8}{3})}\,\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}\,g\right) \right\| _{L^{2}}^{2}, \qquad s=-\frac{3\gamma }{10-3q}. \end{aligned}$$

Using then (4.2), for any $q >2$, one has $\Vert f_{\ell }^{+}\Vert _{L^{2}}^{2} \leqslant (\ell -k)^{2-q}\,\Vert f_{k}^{+}\Vert _{L^{q}}^{q}$ for $0 \leqslant k < \ell$, and the above inequality gives the result.$\square$

Let us now introduce, for any measurable $f:=f(t,v) \geqslant 0$ and $\ell \geqslant 0$, the energy functional

$$\begin{aligned}&{\mathscr {E}}_{\ell }(T_{1},T_{2})=\sup _{t \in [T_{1},T_{2})}\left( \frac{1}{2} \left\| f_{\ell }^{+}(t)\right\| _{L^{2}}^{2} + c_{0}\int _{T_{1}}^{t}\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}\,f^{+}_{\ell }(\tau )\right) \right\| _{L^{2}}^{2}\mathrm {d}\tau \right) ,\\&\qquad 0 \leqslant T_{1} \leqslant T_{2}. \end{aligned}$$

We have then the fundamental result for the implementation of the level set analysis.

Proposition 4.3

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, for any $p_{\gamma } \in (1,3)$ and any $s > \frac{3}{2}|\gamma |$, there exist some positive constants $C_{1},C_{2}$ depending only on s, $\Vert f_{\mathrm{in}}\Vert _{L^1_2}$ and $H(f_{\mathrm{in}})$ such that, for any times $0 \leqslant T_{1} < T_{2} \leqslant T_{3}$ and $0 \leqslant k < \ell$,

$$\begin{aligned} {\mathscr {E}}_{\ell }(T_{2},T_{3})\leqslant & {} \frac{{C}_{2}}{T_{2}-T_{1}}(\ell -k)^{{-\frac{4s+3\gamma }{3s}}}\left[ \sup _{\tau \in [T_{1},T_{3}]}\varvec{m}_{s}(\tau )\right] ^{{\frac{|\gamma |}{s}}}\,{\mathscr {E} }_{k}(T_{1},T_{3})^{{\frac{5s+3\gamma }{3s}}}\nonumber \\&+{C}_{1}\left( {\mathscr {E}}_{k}(T_{1},T_{3})\right) ^{\frac{1}{p_{\gamma }} +\frac{2}{3}}\left( \ell -k\right) ^{-\frac{2}{p_{\gamma }}-\frac{1}{3}}\nonumber \\&\times \left( \ell +\left[ {(\ell -k)^{\frac{2}{p_{\gamma }}-1}+\ell (\ell -k)^{\frac{2}{p_{\gamma }} -2}}\right] {\mathscr {E}}_{k}(T_{1},T_{3})^{1-\frac{1}{p_{\gamma }}}\right) \,. \end{aligned}$$

(4.8)

Proof

Let us fix $0 \leqslant T_{1} < T_{2} \leqslant T_{3}$. Integrating inequality (4.1) over $(t_{1},t_{2})$, we obtain that

$$\begin{aligned}&\frac{1}{2}\Vert f_{\ell }^{+}(t_{2})\Vert _{L^{2}}^{2} + c_{0}\int _{t_{1}}^{t_{2}} \big \Vert \nabla \big (\langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\big ) \big \Vert ^{2}_{L^{2}} \mathrm {d}\tau \leqslant \frac{1}{2}\Vert f_{\ell }^{+}(t_{1})\Vert _{L^{2}}^{2} \\&\qquad + C_{0}\int _{t_{1}}^{t_{2}}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\Vert _{L^{2}}^{2}\mathrm {d}\tau - \ell \int _{t_{1}}^{t_{2}}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f](\tau ,v)\,f_{\ell }^{+}(\tau ,v)\,\mathrm {d}v. \end{aligned}$$

Thus, if $T_{1}\leqslant t_{1} \leqslant T_{2} \leqslant t_{2}\leqslant T_{3}$, one first notices that the above inequality implies that

$$\begin{aligned}&\frac{1}{2}\Vert f_{\ell }^{+}(t_{2})\Vert _{L^{2}}^{2} + c_{0}\int _{T_{2}}^{t_{2}} \big \Vert \nabla \big (\langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\big ) \big \Vert _{L^{2}}^{2} \mathrm {d}\tau \leqslant \frac{1}{2}\Vert f_{\ell }^{+}(t_{1})\Vert _{L^{2}}^{2} \\&\qquad + C_{0}\int _{T_{1}}^{t_{2}}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\Vert _{L^{2}}^{2}\mathrm {d}\tau - \ell \int _{T_{1}}^{t_{2}}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f](\tau ,v)\,f_{\ell }^{+}(\tau ,v)\,\mathrm {d}v, \end{aligned}$$

and, taking the supremum over $t_{2} \in [T_{2},T_{3}]$, one gets

$$\begin{aligned} {\mathscr {E}}_{\ell }(T_{2},T_{3})\leqslant & {} \frac{1}{2}\Vert f_{\ell }^{+}(t_{1})\Vert _{L^{2}}^{2}+ C_{0}\int _{T_{1}}^{T_{3}}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\Vert _{L^{2}}^{2}\mathrm {d}\tau \\&- \ell \int _{T_{1}}^{T_{3}}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f](\tau ,v)\,f_{\ell }^{+}(\tau ,v)\mathrm {d}v, \qquad \forall t_{1} \in [T_{1},T_{2}]. \end{aligned}$$

Integrating now this inequality with respect to $t_{1} \in [T_{1},T_{2}]$, one obtains

$$\begin{aligned} {\mathscr {E}}_{\ell }(T_{2},T_{3})\leqslant & {} \frac{1}{2(T_{2}-T_{1})}\int _{T_{1}}^{T_{2}}\Vert f_{\ell }^{+}(t_{1})\Vert _{L^{2}}^{2}\mathrm {d}t_{1}+ C_{0}\int _{T_{1}}^{T_{3}}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\Vert _{L^{2}}^{2}\mathrm {d}\tau \\&- \ell \int _{T_{1}}^{T_{3}}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\varvec{c}_{\gamma }[f](\tau ,v) \,f_{\ell }^{+}(\tau ,v)\,\mathrm {d}v. \end{aligned}$$

Therefore, applying Proposition 2.4 with $\lambda =\gamma <0$, $g=f$ and $\varphi = f^{+}_{\ell }$, we see that

$$\begin{aligned} {\mathscr {E}}_{\ell }(T_{2},T_{3})\leqslant & {} \frac{1}{2(T_{2}-T_{1})}\int _{T_{1}}^{T_{3}}\Vert f_{\ell }^{+}(\tau )\Vert _{L^{2}}^{2}\,\mathrm {d}\tau + C_{0}\int _{T_{1}}^{T_{3}}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\Vert _{L^{2}}^{2}\,\mathrm {d}\tau \nonumber \\&+ \ell \, C_{\gamma ,p_{\gamma }}(f_{\mathrm {in}})\,\int _{T_{1}}^{T_{3}} \Vert \langle \cdot \rangle ^{\gamma }f_{\ell }^{+}(\tau )\Vert _{L^{1}} \,\mathrm {d}\tau \nonumber \\&+ \ell \, C_{\gamma ,p_{\gamma }}(f_{\mathrm {in}})\,\int _{T_{1}}^{T_{3}} \Vert \langle \cdot \rangle ^{\gamma }f_{\ell }^{+}(\tau )\Vert _{L^{p_{\gamma }}}\, \mathrm {d}\tau , \end{aligned}$$

(4.9)

for $p_{\gamma } >1$ such that $-\gamma \,q_{\gamma } < 3$, where $\frac{1}{p_{\gamma }}+\frac{1}{q_{\gamma }}=1.$ Notice that, since $\gamma \in (-2,0)$, any $p_{\gamma } \in (1,3)$ is admissible. We resort now to Lemma 4.2 to estimate the last three terms on the right-hand side of (4.9). Applying (4.3), one first has

$$\begin{aligned} \begin{aligned} \int _{T_{1}}^{T_{3}}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\Vert _{L^{2}}^{2}\,\mathrm {d}\tau&\leqslant C\,(\ell -k)^{-\frac{4}{3}}\int _{T_{1}}^{T_{3}}\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{k}^{+}(\tau )\right) \right\| _{L^{2}}^{2}\,\left\| f^{+}_{k}(\tau )\right\| _{L^{2}}^{\frac{4}{3}}\mathrm {d}\tau \\&\leqslant \frac{C}{(\ell - k)^{\frac{4}{3}}}\sup _{t\in [T_{1},T_{3}]}\Vert f^{+}_{k}(t) \Vert ^{\frac{4}{3}}_{L^{2}}\int _{T_{1}}^{T_{3}}\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}f^{+}_{k}(\tau )\right) \right\| ^{2}_{L^{2}}\,\mathrm {d}\tau . \end{aligned} \end{aligned}$$

Since

$$\begin{aligned}&\sup _{t\in [T_{1},T_{3}]}\Vert f_{k}^{+}(t)\Vert _{L^{2}}^{\frac{4}{3}} \leqslant \left( 2{\mathscr {E}}_{k}(T_{1},T_{3})\right) ^{\frac{2}{3}} \quad \text { and }\\&\quad \int _{T_{1}}^{T_{3}}\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}f^{+}_{k}(\tau )\right) \right\| ^{2}_{L^{2}}\,\mathrm {d}\tau \leqslant c_{0}^{-1}{\mathscr {E}}_{k}(T_{1},T_{3}), \end{aligned}$$

by definition of the energy functional, we get

$$\begin{aligned} C_{0}\int _{T_{1}}^{T_{3}}\Vert \langle \cdot \rangle ^{\frac{\gamma }{2}}f_{\ell }^{+}(\tau )\Vert _{L^{2}}^{2}\mathrm {d}\tau \leqslant {\bar{C}}_{0}\,(\ell -k)^{-\frac{4}{3}}{\mathscr {E}}_{k}(T_{1},T_{3})^{\frac{5}{3}}, \end{aligned}$$

(4.10)

for some positive constant ${\bar{C}}_{0}$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$. Similarly, using (4.4) first with $p=1$ and then with $p=p_{\gamma }>1$, one deduces that

$$\begin{aligned} \begin{aligned} C_{\gamma ,p_{\gamma }}(f_{\mathrm {in}})\,\int _{T_{1}}^{T_{3}} \Vert \langle \cdot \rangle ^{\gamma }f_{\ell }^{+}(\tau )\Vert _{L^{1}} \mathrm {d}\tau&\leqslant {\bar{C}}_{0}(\ell -k)^{-\frac{7}{3}}{\mathcal {E}}_{k}(T_{1},T_{3})^{\frac{5}{3}}\,,\\ C_{\gamma ,p_{\gamma }}(f_{\mathrm {in}})\,\int _{T_{1}}^{T_{3}} \Vert \langle \cdot \rangle ^{\gamma }f_{\ell }^{+}(\tau )\Vert _{L^{p_{\gamma }}} \mathrm {d}\tau&\leqslant {\bar{C}}_{0}(\ell -k)^{-(\frac{2}{p_{\gamma }}+\frac{1}{3})}{\mathcal {E}}_{k} (T_{1},T_{3})^{\frac{1}{p_{\gamma }}+\frac{2}{3}}\,. \end{aligned} \end{aligned}$$

(4.11)

Regarding the first term in the right-hand side of (4.9), one uses (4.5) with $q=\frac{10}{3}+\frac{\gamma }{s} \in \left( \frac{8}{3},\frac{10}{3}\right)$, where $s >\frac{3}{2}|\gamma |$ is given, to get

$$\begin{aligned} \int _{T_{1}}^{T_{3}}\Vert f^{+}_{\ell }(\tau )\Vert _{L^{2}}^{2}\mathrm {d}\tau\leqslant & {} \frac{c_{q}}{(\ell -k)^{q-2}}\sup _{\tau \in [T_{1},T_{3}]}\Vert \langle \cdot \rangle ^{s}\,f_{k}^{+}(\tau )\Vert _{L^{1}}^{\frac{10}{3}-q}\,\times \\&\times \int _{T_{1}}^{T_{3}}\Vert f_{k}^{+}(\tau )\Vert _{L^{2}}^{2(q-\frac{8}{3})}\left\| \nabla \left( \langle \cdot \rangle ^{\frac{\gamma }{2}}\,f_{k}^{+}(\tau )\right) \right\| _{L^{2}}^{2}\mathrm {d}\tau \\\leqslant & {} \frac{\varvec{c}_{q}}{(\ell -k)^{q-2}}\sup _{\tau \in [T_{1},T_{3}]}\Vert \langle \cdot \rangle ^{s}\, f_{k}^{+}(\tau )\Vert _{L^{1}}^{\frac{10}{3}-q}{\mathscr {E}}_{k}(T_{1},T_{3})^{q-\frac{5}{3}}, \end{aligned}$$

for some positive constant $\varvec{c}_{q} >0.$ Thus

$$\begin{aligned} \int _{T_{1}}^{T_{3}}\Vert f^{+}_{\ell }(\tau )\Vert _{L^{2}}^{2}\mathrm {d}\tau \leqslant \frac{\varvec{c}_{q}}{(\ell -k)^{q-2}}\left( \sup _{\tau \in [T_{1},T_{3}]}\varvec{m}_{s}(\tau )\right) ^{\frac{10}{3}-q}\,{\mathscr {E}}_{k}(T_{1},T_{3})^{q-\frac{5}{3}}.\nonumber \\ \end{aligned}$$

(4.12)

Gathering (4.9)–(4.11)–(4.10)–(4.12) gives the result recalling that $q=\frac{10}{3}+\frac{\gamma }{s}$. $\square$

Remark 4.4

Notice that, for $-\frac{4}{3}< \gamma < 0$, then, one can choose $s=2 > \frac{3}{2}|\gamma |$ in (4.8) to get $\sup _{\tau \in [T_{1},T_{3}]}\varvec{m}_{s}(\tau )=\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$. For $\gamma \leqslant -\frac{4}{3}$, we will rather use (4.8) with the choice $s=3$.

With this, we can implement the level set iteration to deduce Theorem 1.10.

Proof of Theorem 1.10

We first start with short times, that is, we are concerned at this point with the appearance of the norm. In all the proof, $C(f_{\mathrm {in}})$ will denote a generic constant depending only on $f_{\mathrm {in}}$ through its $L^{1}_{2}$-norm and entropy $H(f_{\mathrm{in}})$. Let us fix $T>t_{*}>0$ and let $K>0$ (to be chosen sufficiently large). We consider the sequence of levels and times

$$\begin{aligned} \ell _{n}=K\,\left( 1-\frac{1}{2^{n}}\right) , \qquad t_{n}:=t_{*}\left( 1-\frac{1}{2^{n+1}}\right) , \qquad T> t_{*} >0, \qquad n \in {\mathbb {N}}. \end{aligned}$$

We apply Proposition 4.3 with $T_{3}=T$ and the choices

$$\begin{aligned} k= \ell _{n}< \ell _{n+1}=\ell \,,\quad \quad T_{1}=t_{n}<t_{n+1} =T_{2}\,,\qquad {E}_{n}:={\mathscr {E}}_{\ell _{n}}(t_{n},T), \end{aligned}$$

to conclude that

$$\begin{aligned} {E}_{n+1}\leqslant & {} 2^{{\frac{10s+3\gamma }{3s}}}C_{2}\frac{\varvec{y}_{s}^{{ \frac{|\gamma |}{s}}}}{K^{{\frac{4s+3\gamma }{3s}}}\,t_{*}}\, 2^{n{\frac{7s+3\gamma }{3s}}}{E}_{n}^{{\frac{5s+3\gamma }{3s}}} +\frac{C_{1}}{K^{\frac{2}{p_{\gamma }}-\frac{2}{3}}} \,{E}_{n}^{\frac{1}{p_{\gamma }}+\frac{2}{3}}\,2^{\frac{6+p_{\gamma }}{3p_{\gamma }}(n+1)}\\&+{\frac{C_{1}}{K^{\frac{4}{3}}}\,E_{n}^{\frac{5}{3}} \,2^{\frac{7}{3}(n+1)}(1+2^{-(n+1)})}\,, \end{aligned}$$

that is,

$$\begin{aligned} {E}_{n+1}\leqslant & {} 2^{{\frac{10s+3\gamma }{3s}}}C_{2}\frac{\varvec{y}_{s}^{{\frac{|\gamma |}{s}}}}{K^{{\frac{4s+3\gamma }{3s}}}\,t_{*}}\,2^{n{\frac{7s+3\gamma }{3s}}} {E}_{n}^{{\frac{5s+3\gamma }{3s}}}+\frac{C_{1}}{K^{\frac{2}{p_{\gamma }} -\frac{2}{3}}}\,E_{n}^{\frac{1}{p_{\gamma }}+\frac{2}{3}}\, 2^{\frac{6+p_{\gamma }}{3p_{\gamma }}(n+1)} \nonumber \\&+ \frac{2C_{1}}{K^{\frac{4}{3}}} \,E_{n}^{\frac{5}{3}}\,2^{\frac{7}{3}(n+1)}, \end{aligned}$$

(4.13)

for some positive constants $C_{1},C_{2}$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ (but not on n), where

$$\begin{aligned} \varvec{y}_{s}=\sup _{t \in [0,T)}\varvec{m}_{s}(t). \end{aligned}$$

Notice that

so that Proposition 3.15 together with Corollary 3.10 ensure that

$$\begin{aligned} {E}_0 \leqslant C(f_{\mathrm {in}})\big ( T-\tfrac{ t_{*} }{2} + {t_{*}^{-\frac{3}{2}}}\big )\,. \end{aligned}$$

We look now for a choice of the parameters K and $Q >0$ ensuring that the sequence $(E_{n}^{\star })_{n}$ defined by

$$\begin{aligned} {E}^{\star }_{n}:={E}_0\,Q^{-n}, \qquad n \in {\mathbb {N}}\,, \end{aligned}$$

satisfies (4.13) with the reversed inequality. Notice that

$$\begin{aligned} E_{n+1}^{\star }\geqslant & {} 2^{{\frac{10s+3\gamma }{3s}}}C_{2}\frac{\varvec{y}_{s}^{{\frac{|\gamma |}{s}}}}{K^{{\frac{4s+3\gamma }{3s}}}\,t_{*}}\,2^{n{\frac{7s+3\gamma }{3s}}} \left( E_{n}^{\star }\right) ^{{\frac{5s+3\gamma }{3s}}}\nonumber \\&+\frac{C_{1}}{K^{\frac{2}{p_{\gamma }}-\frac{2}{3}}}\,\left( E_{n}^{\star } \right) ^{\frac{1}{p_{\gamma }}+\frac{2}{3}}\,2^{\frac{6+p_{\gamma }}{3p_{\gamma }}(n+1)}+ \frac{2C_{1}}{K^{\frac{4}{3}}}\,\left( E_{n}^{\star }\right) ^{\frac{5}{3}} \,2^{\frac{7}{3}(n+1)} \end{aligned}$$

(4.14)

is equivalent to

$$\begin{aligned} 1\geqslant & {} \frac{2^{{\frac{10s+3\gamma }{3s}}}\,C_{2}}{K^{ {\frac{4s+3\gamma }{3s}}}\,t_{*}}\varvec{y}_{s}^{ {\frac{|\gamma |}{s}}}\,Q {E}_{0}^{ {\frac{2s+3\gamma }{3s}}}\,\left[ Q^{ {-\frac{2s+3\gamma }{3s}}}\,2^{ {\frac{7s+3\gamma }{3s}}}\right] ^{n}\\&+\frac{2^{\frac{6+p_{\gamma }}{3p_{\gamma }}} C_{1}}{K^{\frac{2}{p_{\gamma }}-\frac{2}{3}}}\,Q\,{E}_{0}^{\frac{1}{p_{\gamma }} -\frac{1}{3}}\left[ Q^{\frac{1}{3}-\frac{1}{p_{\gamma }}}\,2^{\frac{6+p_{\gamma }}{3p_{\gamma }}}\right] ^{n}\\&+\frac{2^{\frac{10}{3}}\,C_{1}}{K^{\frac{4}{3}}}\,Q\,{E}_{0}^{\frac{2}{3}} \left[ 2^{\frac{7}{3}}Q^{-\frac{2}{3}}\right] ^{n}. \end{aligned}$$

We first choose Q in a such a way that all the terms $\left[ \cdots \right] ^{n}$ are smaller than one, i.e.

$$\begin{aligned} Q=\max \left( 2^{\frac{7}{2}},2^{\frac{6+p_{\gamma }}{3-p_{\gamma }}},2^{{\frac{7s+3\gamma }{2s+3\gamma }}}\right) , \end{aligned}$$

where we recall that ${s>\frac{3}{2}|\gamma |}$ and $p_{\gamma } <3.$ With such a choice, (4.14) would hold as soon as

$$\begin{aligned} 1 \geqslant \frac{2^{{\frac{10s+3\gamma }{3s}}}\,C_{2}}{K^{{\frac{4s+3\gamma }{3s}}} \,t_{*}}\varvec{y}_{s}^{{\frac{|\gamma |}{s}}}\,Q {E}_{0}^{{\frac{2s+3\gamma }{3s}}} \,+\frac{2^{\frac{6+p_{\gamma }}{3p_{\gamma }}}C_{1}}{K^{\frac{2}{p_{\gamma }} -\frac{2}{3}}}\,Q\,{E}_{0}^{\frac{1}{p_{\gamma }}-\frac{1}{3}} +\frac{2^{\frac{10}{3}}\,C_{1}}{K^{\frac{4}{3}}}\,Q\,{E}_{0}^{\frac{2}{3}}.\qquad \end{aligned}$$

(4.15)

This would hold for instance if each term of the sum is smaller than $\frac{1}{3}$, and a direct computation shows that this amounts to choose

$$\begin{aligned} K \geqslant K(t_{*},T)=\max \left\{ K_{1}(t_{*},T),K_{2}(t_{*}, T),K_{3}(t_{*},T),K_{4}(t_{*},T)\right\} \end{aligned}$$

with

$$\begin{aligned} {\left\{ \begin{array}{ll} K_{1}(t_{*},T)=1, \qquad K_{2}(t_{*},T)=\left( 3C_{2}{E}_{0}^{{\frac{2s+3\gamma }{3s}}}2^{{\frac{10s+3\gamma }{3s}}}\,Q\,t_{*}^{-1}\varvec{y}_{s}^{-{\frac{\gamma }{s}}}\right) ^{{\frac{3s}{4s+3\gamma }}}\,,\\ K_{3}(t_{*},T)=\left( 3C_{1} {E}_{0}^{\frac{3-p_{\gamma }}{3p_{\gamma }}}2^{\frac{6+p_{\gamma }}{3p_{\gamma }}}Q\right) ^{\frac{3p_{\gamma }}{6-2p_{\gamma }}},\qquad K_{4}(t_{*},T)=\left( 3C_{1} {E}_{0}^{\frac{2}{3}}2^{\frac{10}{3}}Q\right) ^{\frac{3}{4}}\,.\end{array}\right. } \end{aligned}$$

(4.16)

By a comparison principle (because ${E}_{0} ={E}^{\star }_{0}$), one concludes that

$$\begin{aligned} E_{n} \leqslant E_{n}^{\star }\,, \qquad n \in {\mathbb {N}}\,, \end{aligned}$$

and in particular, since $Q >1$,

$$\begin{aligned} \lim _{n}E_{n}=0. \end{aligned}$$

Since $\lim _{n}t_{n}=t_{*}$ and $\lim _{n}\ell _{n}=K$, this implies that

$$\begin{aligned} \sup _{t\in [t_*,T)}\Vert f^{+}_{K}(t)\Vert _{L^{2}}= 0\,, \end{aligned}$$

for $K \geqslant K(t_{*},T)$ and, in particular,

$$\begin{aligned} \Vert f(t)\Vert _{L^{\infty }} \leqslant K(t_*,T)\,,\qquad 0<t_{*}\leqslant t < T\,. \end{aligned}$$

Recall that $K(t_{*},T)=\max \{K_{i}(t_{*},T),i=1,2,3,4\}$ as defined in (4.16). We estimate it roughly by the sum of these four terms, i.e.

$$\begin{aligned} K(t_{*},T) \leqslant 1+\sum _{i=2}^{4}K_{i}(t_{*},T)\,, \end{aligned}$$

and notice that the dependence with respect to $T,t_{*}$ and t is encapsulated in the term $E_{0}$ and $K_{2}(t_{*},T)$ (through $t_{*}^{-1}$ and $\varvec{y}_{s}$). One has easily

$$\begin{aligned} K_{2}(t_{*},T) \leqslant c_{2} E_{0}^{{\frac{2s+3\gamma }{4s+3\gamma }}}t_{*}^{ -{\frac{3s}{4s+3\gamma }}}\varvec{y}_{s}^{{\frac{3|\gamma |}{4s+3\gamma }}}, \qquad K_{3}(t_{*},T) + K_{4}(t_{*},T) \leqslant c_{3} E_{0}^{\frac{1}{2}}\,, \end{aligned}$$

for some positive constants $c_{2},c_{3} >0$ depending on $Q,s,p_{\gamma }$ and $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}, H(f_{\mathrm{in}})$ (through $C_{1},C_{2}$).

Noticing that $E_{0}$ is bounded away from zero (by some constant independent of $t_{*},T$)¹ and that ${E}_0\leqslant C(f_{\mathrm {in}})\big (T-\frac{t_{*}}{2} + {t_{*}^{-\frac{3}{2}}}\big )$, since ${\frac{2s+3\gamma }{4s+3\gamma }} <\frac{1}{2}$, we can derive the estimate

$$\begin{aligned} K(t_{*},T) \leqslant C\left( 1+t_{*}^{-{\frac{3s}{4s+3\gamma }}}\right) \sqrt{ T - \tfrac{t_{*}}{2} + {t_{*}^{-\frac{3}{2}}} }\,\varvec{y}_{s}^{ {\frac{3|\gamma |}{4s+3\gamma }}}\,, \end{aligned}$$

for some positive constant C depending on $Q,s,p_{\gamma }$ and the constants $C_{1},C_{2}$ appearing in (4.16). Thus, taking $0<t_{*}<T=2$, we obtain the result in the time interval (0, 2].

For $T\geqslant 2$, we copycat the previous argument with the increasing sequence of times

$$\begin{aligned} 0 < T - \tfrac{3}{2}=t_{0} \leqslant t_{n} = T - 1 - \frac{1}{2^{n+1}}\,,\qquad n\in {\mathbb {N}}\,. \end{aligned}$$

In this case the first term in the right-hand side of (4.13) can be replaced with (since no dependence upon $t_{*}$ appears)

$$\begin{aligned} 2^{{\frac{10s+3\gamma }{3s}}}C_{2}\frac{\varvec{y}_{s}^{{\frac{|\gamma |}{s}}}}{K^{{\frac{4s+3\gamma }{3s}}}}\,2^{n{\frac{7s+3\gamma }{3s}}}{E}_{n}^{{\frac{5s+3\gamma }{3s}}}\,. \end{aligned}$$

Furthermore, $\lim _{n}t_{n}=T-1$ and, by Corollary  3.10,

$$\begin{aligned} {E}_0\leqslant C(f_{\mathrm {in}})\big (T - t_{0} +1)=C(f_{\mathrm {in}})\big ( T - \big ( T - \tfrac{3}{2} \big )+1\big ) = \tfrac{5}{2}C(f_{\mathrm {in}})\,. \end{aligned}$$

Consequently,

$$\begin{aligned} \sup _{\tau \in [T-1,T]}\Vert f(\tau )\Vert _{L^{\infty }} \leqslant K\leqslant C(f_{\mathrm {in}}) \varvec{y}_{s}^{{\frac{3|\gamma |}{4s+3\gamma }}} \,. \end{aligned}$$

The result follows since $T\geqslant 2$ is arbitrary. $\square$

A simple consequence of the above is the following:

Corollary 4.5

Assume that $-\frac{4}{3}< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, there is a constant $C>0$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that, for any $t_{*} >0$,

$$\begin{aligned} \sup _{t \geqslant t_{*}}\left\| f(t)\right\| _{L^{\infty }} \leqslant C\,\big ( 1+ {t_{*}^{-\frac{6}{8+3\gamma }-\frac{3}{4}}} \big )\,. \end{aligned}$$

(4.17)

In particular, there exists some explicit $\varvec{\varepsilon }^{\dagger }$ and $\kappa _{0}$ both depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that, for any $\varvec{\varepsilon }\in [0,\varvec{\varepsilon }^{\dagger }]$,

$$\begin{aligned} \inf _{v \in {\mathbb {R}}^{3}}\left( 1-\varvec{\varepsilon }f(t,v)\right) \geqslant \kappa _{0} >0, \qquad t \geqslant 1. \end{aligned}$$

(4.18)

Proof

The proof is a direct consequence of Theorem 1.10 (cf. also Remark 4.4) since, for $-\frac{4}{3}< \gamma <0$, we can pick $s=2$ and $\sup _{t \in [0,T]}\varvec{m}_{s}(t)=\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ is independent of T. $\square$

5 Long-time Behaviour: Algebraic Convergence Result

We investigate now the long-time behaviour of solutions to (1.11.10). Our approach is based upon the entropy/entropy dissipation method.

5.1 General strategy and estimates

In this section, for any $\eta \in {\mathbb {R}}$, we will denote by ${\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(g)$ the entropy production associated to the interaction kernel $\Psi (z)=|z|^{\eta +2}$, i.e.

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(g):=\frac{1}{2}\int \int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}} \,|v-v_{*}|^{\eta +2}\varvec{\** }_{\varvec{\varepsilon }}[g](v,v_{*})\,\mathrm {d}v\mathrm {d}v_{*}\,, \end{aligned}$$

(5.1)

where $\varvec{\** }_{\varvec{\varepsilon }}[g](v,v_{*})$ is defined by (2.5). We recall the following result from a previous contribution [15].

Theorem 5.1

Assume that $0 \leqslant g \leqslant \varvec{\varepsilon }^{-1}$ is such that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}} g(v)\,\,\mathrm {d}v = 1, \quad \int _{{\mathbb {R}}^{3}} g(v)\,v_i\,\,\mathrm {d}v = 0 \quad (i=1,2,3)\,, \quad \int _{{\mathbb {R}}^{3}} g(v)\,|v|^2\,\,\mathrm {d}v = 3\,,\nonumber \\ \end{aligned}$$

(5.2)

and let

$$\begin{aligned} \kappa _{0} := \kappa _{0}(g)=\inf _{v\in {\mathbb {R}}^{3}}(1-\varvec{\varepsilon }\,g(v)) >0. \end{aligned}$$

(5.3)

For any $\eta \geqslant 0$,

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(g) \geqslant 2\lambda _{\eta }(g)\left[ b_{\varvec{\varepsilon }}-\frac{12\varvec{\varepsilon }^{2}}{\kappa _{0}^{4}} \max (\Vert g\Vert _{\infty }^{2},\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{\infty }^{2})\right] {\mathcal {H}}_{\varvec{\varepsilon }}(g|{\mathcal {M}}_{\varvec{\varepsilon }}), \end{aligned}$$

where $\lambda _{\eta }(g) >0$ is given by

$$\begin{aligned} \frac{1}{\lambda _{\eta }(g)}:= {510}\frac{\varvec{e}_{g}^{3}\,}{\kappa _{0}^{2}}\, \max (1,B_{g})\,\max \left( 1,\varvec{m}_{2+\eta }(g)\right) {\mathscr {I}}_{\eta }(g)\,, \end{aligned}$$

(5.4)

with

$$\begin{aligned} {\mathscr {I}}_{\eta }(g)=\sup _{v \in {\mathbb {R}}^{3}}\langle v\rangle ^{\eta } \int _{{\mathbb {R}}^{3}}g(w)|w-v|^{-\eta }\langle w\rangle ^{2}\mathrm {d}w , \end{aligned}$$

and

$$\begin{aligned} \frac{1}{B_{g}}:=\min _{i\ne j}\inf _{\sigma \in {\mathbb {S}}^{1}}\int _{{\mathbb {R}}^{3}} \left| \sigma _{1}\frac{v_{i}}{\langle v\rangle }-\sigma _{2} \frac{v_{j}}{\langle v\rangle }\right| ^{2}g(v)\,\mathrm {d}v, \qquad \frac{1}{\varvec{e}_{g}}=\min _{i}\tfrac{1}{3}\int _{{\mathbb {R}}^{3}} g(v)\,v_{i}^2\,\,\mathrm {d}v\,. \end{aligned}$$

Recall that ${\mathcal {M}}_{\varvec{\varepsilon }}$ and $b_{\varvec{\varepsilon }}$ are introduced in Definition 1.1.

Our approach is based on the interpolation between the entropy production with parameter $\gamma$ and the entropy production with parameter $\eta \geqslant 0.$ Namely, for a given g satisfying (1.85.2), a simple consequence of Hölder’s inequality is that

$$\begin{aligned} {\mathscr {D}}^{(0)}_{\varvec{\varepsilon }}(g) \leqslant \Big ( {\mathscr {D}}^{(\gamma )}_{\varvec{\varepsilon }}(g) \Big )^{ \frac{\eta }{\eta -\gamma } }\; \Big ( {\mathscr {D}}^{(\eta )}_{\varvec{\varepsilon }}(g) \Big )^{ \frac{-\gamma }{\eta -\gamma } }\,, \qquad \eta >0\,, \quad \gamma < 0\,, \end{aligned}$$

or equivalently,

$$\begin{aligned} {\mathscr {D}}^{(\gamma )}_{\varvec{\varepsilon }}(g) \geqslant \left( {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(g)\right) ^{1-\frac{\gamma }{\eta }}\, \left( {\mathscr {D}}^{(\eta )}_{\varvec{\varepsilon }}(g)\right) ^{\frac{\gamma }{\eta }}. \end{aligned}$$

(5.5)

Noticing that $1-\frac{\gamma }{\eta } >0$, we can invoke Theorem 5.1 to bound from below ${\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(g)$ in terms of ${\mathcal {H}}_{\varvec{\varepsilon }}(g|{\mathcal {M}}_{\varvec{\varepsilon }})$, and we need to deduce an upper bound for ${\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(g)$. We begin with the lower bound of ${\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t))$ for solutions to (1.11.10), which can be deduced from Theorem 5.1.

Proposition 5.2

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, there exist $\varvec{\varepsilon }_{1} \in (0,\varvec{\varepsilon }_{0}]$ and a positive constant ${\bar{C}}_{1} >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t)) \geqslant {\bar{C}}_{1}\left( 1-98\varvec{\varepsilon }\chi (t)\right) \,{\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})\,, \qquad t\geqslant 0\, ,\qquad {\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{1}]}, \end{aligned}$$

where

$$\begin{aligned} \chi (t):=\max \left( \Vert f(t)\Vert _{L^{\infty }},\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{\infty }}\right) \in \left( 0,\varvec{\varepsilon }^{-1}\right) , \qquad t \geqslant 0. \end{aligned}$$

Proof

From Theorem 5.1, there is some universal constant $c >0$ such that

$$\begin{aligned}&{\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t)) \geqslant 2c\lambda _{0}(t)\left[ b_{\varvec{\varepsilon }}-\frac{12\varvec{\varepsilon }^{2}}{\kappa _{0}(t)^{4}}\max \left( \Vert f(t)\Vert _{L^{\infty }}^{2},\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{\infty }}^{2}\right) \right] {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})\,, \quad t\geqslant 0\,, \end{aligned}$$

with

$$\begin{aligned} \kappa _{0}(t)=\inf _{v \in {\mathbb {R}}^{3}}(1-\varvec{\varepsilon }f(t,v)), \qquad t \geqslant 0\,, \end{aligned}$$

and

$$\begin{aligned} \lambda _{0}(t)^{-1}=\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}^{2} \frac{\varvec{e}(t)^{3}}{\kappa _{0}^{2}(t)}\max (1,B(t))\,. \end{aligned}$$

Here,

$$\begin{aligned}&\frac{1}{B(t)}:=\min _{i\ne j}\inf _{\sigma \in {\mathbb {S}}^{1}}\int _{{\mathbb {R}}^{3}} \left| \sigma _{1}\frac{v_{i}}{\langle v\rangle } -\sigma _{2}\frac{v_{j}}{\langle v\rangle }\right| ^{2}f(t,v) \, \,\mathrm {d}v,\\&\qquad \frac{1}{\varvec{e}(t)}=\min _{i}\tfrac{1}{3}\int _{{\mathbb {R}}^{3}} f(t,v)\,v_{i}^2\,\,\mathrm {d}v, \end{aligned}$$

since

$$\begin{aligned} \max (1,\varvec{m}_{2}(f(t))){\mathscr {I}}_{0}(f(t))=\max (1,\varvec{m}_{2}(t))\, \varvec{m}_{2}(t) \leqslant \Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}^{2}, \qquad t\geqslant 0\,, \end{aligned}$$

by conservation of energy and because $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}} \geqslant 1$. As shown in [15, Remarks 2.10 & 2.11], there is a positive constant $C_{0} >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm {in}})$ such that $\min \left( \frac{1}{B(t)},\frac{1}{\varvec{e}^{3}(t)}\right) \geqslant C_{0}$ for any $t\geqslant 0.$ Therefore, there is a positive constant ${\bar{C}}_{0} >0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm {in}})$ such that

$$\begin{aligned} 2c\lambda _{0}(t) \geqslant C_{0}^{2}\frac{\kappa _{0}(t)^{2}}{\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}^{2}} \geqslant {\bar{C}}_{0}\kappa _{0}(t)^{2}, \qquad t\geqslant 0\,, \end{aligned}$$

and, since $\kappa _{0}(t) \leqslant 1$ and $b_{\varvec{\varepsilon }} \geqslant \frac{1}{8}$ for $\varvec{\varepsilon }$ small enough (see [2,  Lemma A.1]), we easily deduce that

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t)) \geqslant {\bar{C}}_{1}\,\left[ \kappa _{0}(t)^{4}- 96\varvec{\varepsilon }^{2}\,\max \left( \Vert f(t)\Vert _{L^{\infty }}^{2},\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{ L^{\infty }}^{2}\right) \right] {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}),\nonumber \\ \end{aligned}$$

(5.6)

for any $t\geqslant 0$ with ${\bar{C}}_{1}=\tfrac{1}{8}{\bar{C}}_{0}.$ Since $\kappa _{0}(t)=1-\varvec{\varepsilon }\Vert f(t)\Vert _{L^{\infty }}$, one has $\kappa _{0}(t) \geqslant 1-\varvec{\varepsilon }\chi (t)$ for any $t\geqslant 0$ and (5.6) becomes

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t)) \geqslant {\bar{C}}_{1}\,\left[ \left( 1-\varvec{\varepsilon }\chi (t)\right) ^{4}-96\varvec{\varepsilon }^{2}\chi (t)^{2}\right] {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}), \qquad t\geqslant 0. \end{aligned}$$

Expanding $(1-\varvec{\varepsilon }\chi (t))^{4}$ and noticing that $-\varvec{\varepsilon }^{3}\chi ^{3}(t) \geqslant -\varvec{\varepsilon }^{2}\chi ^{2}(t) \geqslant -\varvec{\varepsilon }\chi (t)$ because $\varvec{\varepsilon }\chi (t)\leqslant 1$, one gets the result. $\square$

We now derive an upper bound for ${\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(g)$. A first observation is the following technical estimate:

Lemma 5.3

For any $0 \leqslant g \leqslant \varvec{\varepsilon }^{-1}$ satisfying (5.3) and any $\eta \geqslant -2$, one has

$$\begin{aligned} {\mathscr {D}}^{(\eta )}_{\varvec{\varepsilon }}(g) \leqslant \frac{2^{\frac{\eta +8}{2}}}{\kappa _{0}(g)}\,\Vert g\Vert _{L^{1}_{\eta +2}}\int _{{\mathbb {R}}^{3}}\langle v \rangle ^{\eta +2}\big | \nabla \sqrt{g}\big |^{2}\,\mathrm {d}v\,, \end{aligned}$$

(5.7)

where we recall that $\kappa _{0}(g)=\inf _{v \in {\mathbb {R}}^{3}}\left( 1-\varvec{\varepsilon }g(v)\right) =1-\varvec{\varepsilon }\Vert g\Vert _{L^{\infty }}.$

Proof

Using definition (2.5), one has

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(g)=\frac{1}{2}\int _{{\mathbb {R}}^{6}}|v-v_{*}|^{\eta +2}\,g\,g_{*}(1-\varvec{\varepsilon }g)\,(1-\varvec{\varepsilon }g_{*})\left| \Pi (v-v_{*})\left[ \nabla h-\nabla h_{*}\right] \right| ^{2}\,\mathrm {d}v\mathrm {d}v_{*}, \end{aligned}$$

where $h(v)=\log (g(v))-\log (1-\varvec{\varepsilon }g(v)).$ Using the obvious estimate

$$\begin{aligned} \left| \Pi (v-v_{*})\left[ \nabla h-\nabla h_{*}\right] \right| ^{2} \leqslant 2|\nabla h|^{2}+2|\nabla h_{*}|^{2}\,, \end{aligned}$$

one has

$$\begin{aligned} \begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(g)&\leqslant 2 \int _{{\mathbb {R}}^{6}}|v-v_{*}|^{\eta +2}\,g\,g_{*}(1-\varvec{\varepsilon }g)\,(1-\varvec{\varepsilon }g_{*})\left| \frac{\nabla g(v)}{g(v)(1-\varvec{\varepsilon }g(v))}\right| ^{2}\mathrm {d}v\mathrm {d}v_{*}\\&\leqslant 2\int _{{\mathbb {R}}^{3}}\frac{|\nabla g(v)|^{2}}{g(1-\varvec{\varepsilon }g)}\mathrm {d}v\int _{{\mathbb {R}}^{3}}|v-v_{*}|^{\eta +2}\,g_{*}\mathrm {d}v_{*}. \end{aligned} \end{aligned}$$

Using the fact that $|v-v_{*}|^{\eta +2} \leqslant 2^{\frac{\eta +2}{2}}\langle v\rangle ^{\eta +2}\langle v_{*}\rangle ^{\eta +2}$, we get the desired result thanks to (5.3). $\square$

On the basis of estimates (5.7) and (5.5), we need to provide a uniform in time upper bound of the above weighted Fisher information along solutions to (1.11.10). We follow the approach of [27] and begin with a technical Lemma:

Lemma 5.4

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, for any $s >\frac{3}{2}$, there exists $C_{s}(f_{\mathrm {in}}) >0$ depending on s, $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that, for any $t\geqslant 0$ and $k \geqslant 0$

(5.8)

and

(5.9)

Proof

We use the following obvious estimate: for any $p,r >1$, there is $C_{p,r} >0$ such that

$$\begin{aligned} x\left( 1+|\log x|\right) \leqslant C_{p,r}\left( x^{\frac{1}{r}}+x^{p}\right) \,, \qquad \forall \, x >0. \end{aligned}$$

(5.10)

For notational simplicity, in several places we omit the dependence of f with respect to t. Splitting $\varvec{c}_{\gamma }[f]$ as

$$\begin{aligned} \varvec{c}_{\gamma }[f]=-2(\gamma +3)\left[ \left( |\cdot |^{\gamma }{\mathbf {1}}_{|\cdot | \leqslant 1}*f \right) + \left( |\cdot |^{\gamma }{\mathbf {1}}_{|\cdot | >1} *f\right) \right] =\varvec{c}_{\gamma }^{(1)}[f]+\varvec{c}_{\gamma }^{(2)}[f]\,, \end{aligned}$$

one has that

$$\begin{aligned} -\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }[f]\,f\left( 1+|\log f|\right) \,\mathrm {d}v\leqslant & {} -C_{\frac{4}{3},\frac{3}{2}}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }^{ (1)}[f]\left( f^{\frac{2}{3}}+f^{\frac{4}{3}}\right) \,\mathrm {d}v \\&-C_{2,2}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }^{(2)}[f]\left( \sqrt{f}+f^{2}\right) \,\mathrm {d}v\,. \end{aligned}$$

Clearly,

$$\begin{aligned} -\varvec{c}_{\gamma }^{(2)}[f]=2(\gamma +3)\int _{|v-v_{*}| > 1}|v-v_{*}|^{\gamma }f(v_{*})\mathrm {d}v_{*}\leqslant 2(\gamma +3)\Vert f_{\mathrm{in}}\Vert _{L^{1}}\,, \end{aligned}$$

so that

$$\begin{aligned}&-\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }^{(2)}[f]\left( \sqrt{f}+f^{2}\right) \,\mathrm {d}v \\&\quad \leqslant 2(\gamma +3)\Vert f_{\mathrm{in}}\Vert _{L^{1}} \left( \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\sqrt{f}\mathrm {d}v+\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}f^{2}(v)\,\mathrm {d}v\right) . \end{aligned}$$

According to Cauchy-Schwarz inequality, for any $s >\frac{3}{2}$

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\sqrt{f}\,\mathrm {d}v \leqslant \sqrt{\varvec{m}_{2(k+s)}(f)}\Vert \langle \cdot \rangle ^{-s}\Vert _{L^{2}}=C_{s}\sqrt{\varvec{m}_{2(k+s)}(f)}\,, \end{aligned}$$

and consequently

for some positive constant depending only on s and $\Vert f_{\mathrm{in}}\Vert _{L^{1}}$. Using Hölder’s inequality,

$$\begin{aligned} -\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }^{(1)}[f]\left( f^{\frac{2}{3}}+f^{\frac{4}{3}}\right) \mathrm {d}v \leqslant \left\| \langle \cdot \rangle ^{k}\left( f^{\frac{2}{3}}+f^{\frac{4}{3}}\right) \right\| _{L^{\frac{3}{2}}}\,\left\| \varvec{c}_{\gamma }^{(1)}[f]\right\| _{L^{3}}, \end{aligned}$$

and, according to Young’s convolution inequality,

$$\begin{aligned} \left\| \varvec{c}_{\gamma }^{(1)}[f]\right\| _{L^{3}} \leqslant 2(\gamma +3)\left\| \,|\cdot |^{\gamma }{\mathbf {1}}_{|\cdot |\leqslant 1}\right\| _{L^{\frac{3}{2}}}\,\Vert f\Vert _{L^{\frac{3}{2}}} , \end{aligned}$$

where $\left\| \,|\cdot |^{\gamma }{\mathbf {1}}_{|\cdot |\leqslant 1}\right\| _{L^{\frac{3}{2}}} < \infty$ since $\frac{3}{2}\gamma +3>0$ (recall that $\gamma \in (-2,0))$. Since

$$\begin{aligned} \Vert f\Vert _{L^{\frac{3}{2}}} \leqslant \left( \varvec{m}_{0}(f)+\Vert f\Vert _{L^{2}}^{2}\right) ^{\frac{2}{3}} , \end{aligned}$$

we deduce that

Now using Proposition 3.15, this proves (5.8). Now, by (5.10), one has

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k+\gamma }\,f\left( 1+|\log f|\right) \,\mathrm {d}v \leqslant C_{2,2}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k+\gamma }\left( \sqrt{f}+f^{2}\right) \,\mathrm {d}v, \end{aligned}$$

and, proceeding as above, one obtains (5.9). $\square$

We can state now the following Proposition which is inspired by [27]:

Proposition 5.5

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Let $\eta \geqslant \gamma -2.$ Assume moreover that

$$\begin{aligned} {f_{\mathrm {in}} \in L^{1}_{2{\eta +8-2\gamma }}({\mathbb {R}}^{3})}. \end{aligned}$$

Then, for any $t_{0} >0$, there exists $C >0$ depending on $\eta ,t_{0}$ and on $f_{\mathrm {in}}$ through $\Vert f_{\mathrm {in}}\Vert _{L^1_{2{\eta +8-2\gamma }}}$ such that

$$\begin{aligned} \int _{t_{0}}^{t}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{ \eta +2}\left| \nabla \sqrt{f(\tau ,v)}\right| ^{2}\,\mathrm {d}v \leqslant C(1+t)^{2}\,, \qquad 0\leqslant t_{0} < t. \end{aligned}$$

(5.11)

In particular, for $\eta \geqslant 0$, there is $C_{\eta }(f_{\mathrm {in}})$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ and $\eta$ and such that

$$\begin{aligned} \int _{t_{0}}^{t}{\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(f(\tau ))\mathrm {d}\tau \leqslant C_{\eta }(f_{\mathrm {in}})\left[ \sup _{t_0\leqslant \tau \leqslant t} \frac{\varvec{m}_{\eta +2}(\tau )}{\kappa _{0}(\tau )}\right] \,(1+t)^{2} \,,\qquad 0\leqslant t_{0} < t\,, \end{aligned}$$

(5.12)

where we recall that $\kappa _{0}(\tau )=1-\varvec{\varepsilon }\,\Vert f(\tau )\Vert _{L^{\infty }}$, $\tau \geqslant 0.$

Proof

Let $\eta \geqslant \gamma -2$ be fixed. Since we aim to use Lemma 5.4, it will be convenient here to introduce $k =\eta +2-\gamma$. We compute, as in [27,  Proposition 1] the evolution of

$$\begin{aligned} S_{k}(t):=\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}f(t,v)\log f(t,v)\,\mathrm {d}v, \end{aligned}$$

for a solution $f=f(t,v)$ to (1.11.10). To simplify notations, we write $F=F(t,v)=f(1-\varvec{\varepsilon }f)$. One can check that

$$\begin{aligned} \begin{aligned} \dfrac{\mathrm {d}}{\mathrm {d}t}S_{k}(t)&=\dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{k}(t) +\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k} {\mathcal {Q}}(f)\log f\,\mathrm {d}v \\&=\dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{k}(t)+\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k} \nabla \cdot \left( \varvec{\Sigma }[f]\nabla f\right) \log f\,\mathrm {d}v\\&\quad -\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\nabla \cdot \left( \varvec{b}[f]F\right) \log f\,\mathrm {d}v. \end{aligned} \end{aligned}$$

(5.13)

One computes, using that $\log f \,\nabla f=\nabla \left[ f\log f-f\right]$, that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\nabla \cdot \left( \varvec{\Sigma }[f]\nabla f\right) \log f\mathrm {d}v= & {} -\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{\Sigma }[f]\nabla f\cdot \frac{\nabla f}{f}\,\mathrm {d}v\\&+k\int _{{\mathbb {R}}^{3}}\nabla \cdot \left( \varvec{\Sigma }[f]v \langle v\rangle ^{k-2}\right) \,\left[ f\log f-f\right] \,\mathrm {d}v. \end{aligned}$$

Similarly,

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\nabla \cdot \left( \varvec{b}[f]F\right) \log f\,\mathrm {d}v= & {} -k\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k-2}\left( \varvec{b}[f]\cdot v\right) \,F\log f\,\mathrm {d}v\\&+\int _{{\mathbb {R}}^{3}}\left( f-\frac{\varvec{\varepsilon }}{2}f^{2}\right) \nabla \cdot \left( \langle v\rangle ^{k}\varvec{b}[f]\right) \,\mathrm {d}v. \end{aligned}$$

As in the proof of Lemma 3.8,

$$\begin{aligned} \nabla \cdot \left( \varvec{\Sigma }[f]v \langle v\rangle ^{k-2}\right) =\langle v\rangle ^{k-2}\varvec{B}[f]\cdot v +\langle v\rangle ^{k-4}\mathrm {Trace}\left( \varvec{\Sigma }[f]\cdot \varvec{A}(v)\right) , \end{aligned}$$

with $\varvec{A}(v)=\langle v\rangle ^{2}\mathbf {Id}+(k-2)\,v\otimes v,$ whereas

$$\begin{aligned} \nabla \cdot \left( \langle v\rangle ^{k}\varvec{b}[f]\right) =k\langle v\rangle ^{k-2} \left( \varvec{b}[f]\cdot v\right) + \langle v\rangle ^{k}\varvec{c}_{\gamma }[f], \end{aligned}$$

resulting in

$$\begin{aligned} \begin{aligned} \dfrac{\mathrm {d}}{\mathrm {d}t}S_{k}(t)&=\dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{k}(t)-\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{\Sigma }[f] \nabla f \cdot \frac{\nabla f}{f}\,\mathrm {d}v -\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }[f]\left( f-\frac{\varvec{\varepsilon }}{2}f^{2}\right) \,\mathrm {d}v\\&\quad +k\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k-4}\mathrm {Trace}\left( \varvec{\Sigma }[f]\cdot \varvec{A}(v)\right) \left[ f\log f-f\right] \,\mathrm {d}v\\&\quad +k\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k-2}\left( \varvec{b}[f]\cdot v\right) \left[ F\log f-f+\frac{\varvec{\varepsilon }}{2}f^{2}\right] \,\mathrm {d}v\\&\quad + k \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k-2}\left( \varvec{B}[f]\cdot v\right) \left[ f\log f-f\right] \,\mathrm {d}v. \end{aligned} \end{aligned}$$

(5.14)

From (5.14), using the coercivity of $\varvec{\Sigma }[f]$ and the fact that $-\varvec{c}_{\gamma }[f] \geqslant 0$, we get

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t}S_{k}(t)-\dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{k}(t) +K_{0}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k+\gamma }\frac{|\nabla f|^{2}}{f}\,\mathrm {d}v \nonumber \\&\quad \leqslant {- \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }[f]\,f\,\mathrm {d}v} +k\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k-4}\Big |\mathrm {Trace}\left( \varvec{\Sigma }[f]\cdot \varvec{A}(v)\right) \Big |\left| f\log f-f\right| \,\mathrm {d}v\nonumber \\&\qquad +k\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k-2}\left| \varvec{b}[f]\cdot v\right| \left| F\log f-f+\frac{\varvec{\varepsilon }}{2}f^{2}\right| \,\mathrm {d}v\nonumber \\&\qquad +k\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k-2}\left| \varvec{B}[f]\cdot v\right| \left| f\log f-f\right| \,\mathrm {d}v. \end{aligned}$$

(5.15)

As in the proof of Proposition 3.9, we see that

$$\begin{aligned} \left| \mathrm {Trace}\left( \varvec{\Sigma }[f]\cdot \varvec{A}(v)\right) \right| \leqslant 9\cdot 2k\langle v\rangle ^{2} \left( |\cdot |^{\gamma +2}*f\right) \leqslant 9\cdot 2^{\frac{\gamma +4}{2}}\,k\,\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}\langle v\rangle ^{\gamma +4}, \end{aligned}$$

and, since $|\varvec{B}[f]\cdot v| \leqslant |\varvec{b}[f]\cdot v| +\varvec{\varepsilon }|\varvec{b}[f^2]\cdot v|$ with $\varvec{\varepsilon }f^{2} \leqslant f$, one can check that $\frac{1}{2}|\varvec{B}[f]\cdot v|$ also satisfies (3.44). We deduce then from (5.15) that there exists a constant $C_{k}(f_{\mathrm {in}})>0$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and k, such that

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t}S_{k}(t)-\dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{k}(t) +K_{0}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k+\gamma }\frac{|\nabla f|^{2}}{f}\,\mathrm {d}v \nonumber \\&\quad \leqslant C_{k}(f_{\mathrm {in}})\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k+\gamma }\left( \left| f\log f-f\right| +\left| F\log f-f+\frac{\varvec{\varepsilon }}{2}f^{2}\right| \right) \,\mathrm {d}v\nonumber \\&\qquad -C_{k}(f_{\mathrm {in}})\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }[f]\,\left( \left| f\log f-f\right| +\left| F\log f-f+\frac{\varvec{\varepsilon }}{2}f^{2}\right| {+f}\right) \,\mathrm {d}v.\nonumber \\ \end{aligned}$$

(5.16)

Since

$$\begin{aligned} |f \log f-f|+|F\log f-f+\frac{\varvec{\varepsilon }}{2}f^{2}| \leqslant 2f|\log f|+ \frac{5}{2}f, \end{aligned}$$

(5.17)

we have that

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t}S_{k}(t)-\dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{m}_{k}(t) +K_{0}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k+\gamma }\frac{|\nabla f|^{2}}{f}\,\mathrm {d}v \nonumber \\&\quad \leqslant \frac{5}{2}C_{k}(f_{\mathrm {in}})\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k+\gamma }f\left( 1+\left| \log f\right| \right) \,\mathrm {d}v\nonumber \\&\qquad -\frac{7}{2}C_{k}(f_{\mathrm {in}})\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }[f]\,f\left( 1+\left| \log f\right| \right) \,\mathrm {d}v. \end{aligned}$$

(5.18)

Using Lemma 5.4 with $s=2$, we deduce then that, for any $t_{0} >0$,

where we used Theorem 1.9 for the last estimate and where $C_{k}(f_{\mathrm{in}})$ now depends on $t_{0}$. Notice that, for $s=2k+4 >4+|\gamma |$, Theorem 1.9 provides the growth of $\varvec{m}_{2k+4}(t)$, and $\varvec{E}_{\frac{3k}{2}}(t)$ whenever $\varvec{m}_{s}(0) < \infty$. Our assumption on $f_{\mathrm{in}}$ exactly means that $\varvec{m}_{2k+4}(0)< \infty .$ Integrating this inequality over $(t_{0},t)$ yields

$$\begin{aligned}&K_{0}\int _{t_{0}}^{t}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma +k}\frac{|\nabla {f(\tau ,v)}|^{2}}{f(\tau ,v)}\,\mathrm {d}v\\&\leqslant S_{k}(t_{0})-S_{k}(t)+\varvec{m}_{k}(t) + \frac{1}{2}C_{k}(f_{\mathrm {in}})(1+t)^{2}. \end{aligned}$$

Clearly, $S_{k}(t)$ has no sign but, according to [27,  Eq. (B.3), Lemma B.4], for any $\sigma >0$ there exists $C_{\sigma } >0$ such that

$$\begin{aligned} -S_{k}(t) \leqslant -\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}f(t,v)|\log f(t,v)|\,\mathrm {d}v+2\varvec{m}_{k+\sigma }(t)+C_{\sigma } , \end{aligned}$$

yielding, for $\sigma = 2$,

$$\begin{aligned} K_{0} \int _{{t_0}}^{t}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma +k}|\nabla \sqrt{f(\tau ,v)}|^{2}\,\mathrm {d}v \leqslant C_k(f_{\mathrm{in}})\,(1+t)^{2} {+} S_{k}(t_{0}). \end{aligned}$$

Let us note here that with our assumptions, one can deduce from (5.9) and Theorem 1.9 that $S_{k}(t_{0})$ is actually finite. Indeed, (5.9) implies that $S_{k}(t_{0}) < \infty$ if $\varvec{m}_{2k+2r}(t_{0}) < \infty$ and for some $r > \frac{3}{2}$. According to Theorem 1.9, this holds if $\varvec{m}_{{s}}(0) < \infty$ for $s=2k+2r > 4-\gamma$. As already observed, one has $\varvec{m}_{2k+2r}(0) < \infty$ with $r=2 >\frac{3}{2}$. Recalling that $k+\gamma =\eta +2$, this proves (5.11) with a positive constant C depending in particular on $t_{0}$ (with $C \lesssim t_{0}^{-2}$). We deduce then (5.12) from (5.7) and (5.11). $\square$

5.2 The case $-\frac{4}{3}< \gamma < 0$

We apply the result established here above to the case $\gamma \in \left( -\frac{4}{3},0\right)$. In that case Proposition 5.2 can be stated as:

Proposition 5.6

Assume that $-\frac{4}{3}< \gamma <0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, there exists $\varvec{\varepsilon }^{\star } \in (0,\varvec{\varepsilon }_{0}]$ and ${\bar{\lambda }}_{0} >0$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that, for $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }^{\star }]$,

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t)) \geqslant {\bar{\lambda }}_{0}\, {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})\,, \qquad t\geqslant 1.\end{aligned}$$

(5.19)

Proof

The proof is a direct consequence of Proposition 5.2 and Corollary 4.5 since, recalling that $\sup _{\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]}\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{\infty }}<\infty$ by [2, Lemma A.1], one has

$$\begin{aligned} \chi (t)=\max \left( \Vert f(t)\Vert _{L^{\infty }},\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{\infty }}\right) \leqslant C\,, \qquad t\geqslant 1 , \end{aligned}$$

with $C>0$ independent of $\varvec{\varepsilon }$. Thus, there exists $\varvec{\varepsilon }^{\star } \in (0,\varvec{\varepsilon }^{\dagger })$ such that $\inf _{t\geqslant 1}\left( 1-98\varvec{\varepsilon }\chi (t)\right) > 0$ for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }^{\star }).$ $\square$

Remark 5.7

The restriction $-\frac{4}{3}< \gamma <0$ is needed here only to ensure that $\left( 1-98\varvec{\varepsilon }\chi (t)\right) > 0$. Thus, the above estimate (5.19) holds in any situation for which ${\bar{\lambda }}_0={\bar{C}}_1 \inf _{t\geqslant 1}\left( 1-98\varvec{\varepsilon }\chi (t)\right) >0$.

This gives the following version of Theorem 1.7 where the assumptions on the initial datum are slightly relaxed with respect to the statement of Theorem 1.7:

Proposition 5.8

Let $-\frac{4}{3}< \gamma < 0$. Let $\eta >2 |\gamma |$ and $0\leqslant f_{\mathrm {in}} \in L^{1}_{2\eta +8+2|\gamma |}({\mathbb {R}}^{3})$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, there exists $C_{\eta }$ depending on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ and $\eta >0$, and there exists $\varvec{\varepsilon }^{\ddagger } \in (0,\varvec{\varepsilon }_{0}]$ such that for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }^{\ddagger })$,

$$\begin{aligned} \left\| f(t)-{\mathcal {M}}_{\varvec{\varepsilon }}\right\| _{L^{1}} \leqslant C_{\eta }\,\left( 1+t\right) ^{{-\frac{\eta -2|\gamma |}{2|\gamma |}}} , \qquad \forall t \geqslant 1\,. \end{aligned}$$

(5.20)

As a consequence, given $s > 2|\gamma |$, one has

$$\begin{aligned} \sup _{t\geqslant 1}\varvec{E}_{s}(t) < \infty , \end{aligned}$$

provided that $f_{\mathrm{in}} \in L^{1}_{r}$ with $r > {\max (2s+8+2|\gamma |,\frac{s^{2}}{s-2|\gamma |})}$.

Proof

Using Proposition 5.6 and (5.5), for any $\eta >0$,

$$\begin{aligned} {\mathscr {D}}^{(\gamma )}_{\varvec{\varepsilon }}(f(t)) \geqslant {\bar{\lambda }}_{0}^{1-\frac{\gamma }{\eta }}\,{\mathscr {D}}_{\varvec{\varepsilon }}^{( \eta )}(f(t))^{\frac{\gamma }{\eta }}\,{\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})^{1-\frac{\gamma }{\eta }}, \qquad t\geqslant 1. \end{aligned}$$

For simplicity, we set

$$\begin{aligned} A_{\eta }(t) :={\bar{\lambda }}_{0}^{1-\frac{\gamma }{\eta }}\,{\mathscr {D} }_{\varvec{\varepsilon }}^{(\eta )}(f(t))^{\frac{\gamma }{\eta }} \geqslant 0, \qquad \qquad \varvec{y}(t) :={\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})\,,\qquad t\geqslant 0. \end{aligned}$$

Since $\dfrac{\mathrm {d}}{\mathrm {d}t}{\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})=-{\mathscr {D}}^{(\gamma )}_{\varvec{\varepsilon }}(f(t))$, one has

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\varvec{y}(t) + A_{\eta }(t)\varvec{y}(t)^{1-\frac{\gamma }{\eta }} \leqslant 0, \qquad t\geqslant 1. \end{aligned}$$

Integrating this inequality, we deduce that

$$\begin{aligned} \varvec{y}(t)^{\frac{\gamma }{\eta }}\geqslant \varvec{y}(1)^{\frac{\gamma }{\eta }}-\frac{\gamma }{\eta }\int _{1}^{t}A_{\eta }(\tau )\mathrm {d}\tau \geqslant \varvec{y}(0)^{\frac{\gamma }{\eta }}-\frac{\gamma }{\eta }\int _{1}^{t}A_{\eta }(\tau )\mathrm {d}\tau , \end{aligned}$$

i.e.

$$\begin{aligned} {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}) \leqslant \left( {\mathcal {H}}_{\varvec{\varepsilon }}(f_{\mathrm {in}}|{\mathcal {M}}_{\varvec{\varepsilon }})^{\frac{\gamma }{\eta }} -\frac{\gamma }{\eta }\int _{1}^{t}A_{\eta }(\tau )\mathrm {d}\tau \right) ^{\frac{\eta }{\gamma }}, \qquad t\geqslant 1. \end{aligned}$$

Let us estimate from below the integral of $A_{\eta }(\tau )$. One has

$$\begin{aligned} \begin{aligned} \int _{1}^{t}A_{\eta }(\tau )\,\mathrm {d}\tau&={\bar{\lambda }}_{0}^{1-\frac{\gamma }{\eta }} \int _{1}^{t}{\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(f(\tau ))^{\frac{\gamma }{\eta }}\mathrm {d}\tau ={\bar{\lambda }}_{0}^{1-\frac{\gamma }{\eta }}(t-1)\int _{1}^{t} {\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(f(\tau ))^{\frac{\gamma }{\eta }}\frac{\mathrm {d}\tau }{t-1}\\&\geqslant \bar{\lambda _{0}}^{1-\frac{\gamma }{\eta }}(t-1)\left( \int _{1}^{t} {\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(f(\tau ))\frac{\mathrm {d}\tau }{t-1}\right) ^{\frac{\gamma }{\eta }}, \end{aligned} \end{aligned}$$

where we used Jensen’s inequality and the convexity of the map** $x >0 \mapsto x^{\frac{\gamma }{\eta }}.$ Therefore,

$$\begin{aligned} \int _{1}^{t}A_{\eta }(\tau )\mathrm {d}\tau \geqslant \left( {\bar{\lambda }}_{0}(t-1)\right) ^{1-\frac{\gamma }{\eta }} \left( \int _{1}^{t}{\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(f(\tau ))\mathrm {d}\tau \right) ^{\frac{\gamma }{\eta }}, \end{aligned}$$

and, using (5.12), which holds since $f_{\mathrm{in}} \in L^{1}_{2\eta + 8 - 2 \gamma }({\mathbb {R}}^{3})$, one gets

$$\begin{aligned} \int _{1}^{t}A_{\eta }(\tau )\, \mathrm {d}\tau {\geqslant C}\, (t-1)^{1-\frac{\gamma }{\eta }}{\left( 1+t\right) ^{\frac{3\gamma }{\eta }}} {\geqslant } C {t^{1+\frac{2\gamma }{\eta }}}, \qquad t \geqslant 2, \end{aligned}$$

for some positive constant C depending on $\eta ,\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ where we used (1.14) and the fact that $\inf _{\tau \geqslant 1}\kappa _{0}(\tau ) >0$. Choosing $\eta > -2\gamma$, this gives a decay rate

$$\begin{aligned} {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}) \leqslant C_{\eta }(f_{\mathrm {in}})\,{t^{2+\frac{\eta }{\gamma }}}, \end{aligned}$$

for all $t\geqslant 2$ with $C_{\eta }(f_{\mathrm{in}})$ depending on $\eta ,\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$. We conclude then with Csiszár-Kullback inequality for Fermi-Dirac relative entropy (2.2). Let us prove now the bound on $\varvec{E}_{s}(t)$ for $s \geqslant 0$. It follows from some standard arguments (see [28]). Namely, let $s > -2\gamma$ be given and let $p=\frac{s^{2}}{s+2\gamma } >s$. If $f_{\mathrm{in}} \in L^{1}_{r}$ with $r > \max (2s+8-2\gamma ,p)$, the bound (1.14) in Theorem 1.9 holds as well as the above (5.20) with $\eta =s$. Then, for some positive $C_{s}$ depending only on s, $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$, $H(f_{\mathrm{in}})$ and $\varvec{m}_{r}(0)$, one has

$$\begin{aligned} \begin{aligned} \varvec{m}_{s}(t)&\leqslant \Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{s}}+\Vert f(t)-{\mathcal {M}}_{\varvec{\varepsilon }}(t)\Vert _{L^{1}_{s}}\\&\leqslant \Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{s}}+\Vert f(t)-{\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}}^{1-\theta }\,\Vert f(t) -{\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{p}}^{\theta }\\&\leqslant \Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{s}}+C_{s}(1+t)^{-(1-\theta )\frac{s+2\gamma }{2|\gamma |}}\, \left( \varvec{m}_{p}(t)^{\theta }+\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{p}}^{\theta }\right) \\&\leqslant \Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{s}}+C_{s}(1+t)^{-(1-\theta )\frac{s+2\gamma }{2|\gamma |}} \left( \varvec{C}_{p}^{\theta }\left( 1+t\right) ^{\theta }+\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{p}}^{\theta }\right) ,\\ \theta&=\frac{s+2\gamma }{s} \in (0,1)\,, \end{aligned} \end{aligned}$$

for any $t\geqslant 1$ (so that (5.20) holds). Using that $\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{s}}$ and $\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{p}}$ are bounded uniformly with respect to $\varvec{\varepsilon }$ (see [2,  Lemma A.1]), we deduce that there is $c_{s} >0$ depending only on s, $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $\varvec{m}_{r}(0)$ and $H(f_{\mathrm{in}})$ but not $\varvec{\varepsilon }$ such that

$$\begin{aligned} \varvec{m}_{s}(t) \leqslant c_s \left( 1+\left( 1+t\right) ^{-(1-\theta ) \frac{s+2\gamma }{2|\gamma |}+\theta }\right) \,, \qquad t \geqslant 1. \end{aligned}$$

Since $-(1-\theta )\frac{s+2\gamma }{2|\gamma |}+\theta =0$, this proves that $\sup _{t\geqslant 0}\varvec{m}_{s}(t) \leqslant 2c_{s}$. The proof is similar for the estimate of where we notice that $\max (p,\frac{2p-3\gamma }{4})=p$ which ensures the appearance of the $L^{2}$-moment thanks to Propositions 3.12 and 3.15. The result follows. $\square$

5.3 Proof of Theorem 1.7: the case $-2 < \gamma \leqslant -\frac{4}{3}$

We are in position to give here the full proof of Theorem 1.7. It suffices to consider the case $-2 < \gamma \leqslant -\frac{4}{3}$ since the case $-\frac{4}{3}< \gamma < 0$ has been covered by Proposition 5.8 where in that result, $\eta =\frac{1}{2}\left( s-8+2\gamma \right)$. With respect to the proof of Proposition 5.8, we no longer have a direct control of the norm $\sup _{t\geqslant 1}\Vert f(t)\Vert _{\infty }$.

Recall that, according to Proposition 5.2, there is ${\bar{C}}_{1} >0$ depending on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ such that

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t)) \geqslant {\bar{C}}_{1}\left[ 1-98\,\varvec{\varepsilon }\chi (t)\right] \,{\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}), \qquad t \geqslant 0\,, \end{aligned}$$

with $\chi (t)=\max \Big (\Vert f(t)\Vert _{L^{\infty }}\,,\,\sup _{\varvec{\varepsilon }>0}\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{\infty }}\Big )$. Let us fix $T>2$ and define

$$\begin{aligned} \chi ^{\star }=\chi ^{\star }(T):=98\sup _{t\in [1,T)}\chi (t)\,, \end{aligned}$$

so that

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t)) \geqslant {\bar{C}}_{1}\left( 1-\varvec{\varepsilon }\chi ^{\star }\right) {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})\,, \qquad t \in [1,T). \end{aligned}$$

(5.21)

Pick $\varvec{\varepsilon }:=\varvec{\varepsilon }(T)$ such that

$$\begin{aligned} 1-\varvec{\varepsilon }\chi ^{\star } >0. \end{aligned}$$

Note that the existence of such $\varvec{\varepsilon }$ follows from Theorems 1.9 and 1.10, since ${s >\max (4-\gamma ,-\frac{3}{2}\gamma )}$ with our assumptions. The idea is to couple the a priori estimates with the entropy method to be able to conclude that in fact these quantities are independent of $T>2$ as long as $\varvec{\varepsilon }>0$ is less than some threshold depending only on the initial distribution $f_{\mathrm {in}}$. The interpretation of this condition is that the initial distribution is not too saturated for the argument to hold. It is an open problem to prove that the relaxation to thermal equilibrium happens with a specific rate when $f_{\mathrm {in}}$ is very close to a saturated state even in the hard potential case, see [2].

As in the proof of Proposition 5.8, we write

$$\begin{aligned} \varvec{y}(t)={\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}), \qquad t\geqslant 0. \end{aligned}$$

Recall that we assume here that $f_{\mathrm{in}} \in L^{1}_{s}({\mathbb {R}}^{3})$ with $s > 14+6|\gamma |.$ For notational simplicity, we write $s=2\eta +8-2\gamma$ with $\eta > 3+2|\gamma |.$ In all the sequel, we have then

$$\begin{aligned} f_{\mathrm{in}}\in L^1_{2\eta +8-2\gamma }({\mathbb {R}}^3) \qquad \text { with } \quad \eta >3+2|\gamma |. \end{aligned}$$

(5.22)

Using (5.5), for such a choice of $\eta$, we deduce from (5.21) that

$$\begin{aligned} {\mathscr {D}}^{(\gamma )}_{\varvec{\varepsilon }}(f(t)) \geqslant {\bar{C}}_{1}^{1-\frac{\gamma }{\eta }} \left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{1-\frac{\gamma }{\eta }}\, \left[ {\mathscr {D}}^{(\eta )}_{\varvec{\varepsilon }}(f(t))\right] ^{\frac{\gamma }{\eta }}\, \varvec{y}(t)^{1-\frac{\gamma }{\eta }}, \qquad t \in [1,T). \end{aligned}$$

Recalling that $\dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{y}(t)=-{\mathscr {D}}_{\varvec{\varepsilon }}^{(\gamma )}(f(t))$, we deduce after integration of the above inequality that

$$\begin{aligned} \varvec{y}(t)^{\frac{\gamma }{\eta }}-\varvec{y}(1)^{\frac{\gamma }{\eta }} \geqslant {\bar{C}}(\gamma ,\eta )\left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{1-\frac{\gamma }{\eta }} \int _{1}^{t}\left[ {\mathscr {D}}^{(\eta )}_{\varvec{\varepsilon }}(f(\tau ))\right] ^{\frac{\gamma }{\eta }}\mathrm {d}\tau , \quad t \in [1,T)\,,\nonumber \\ \end{aligned}$$

(5.23)

where we set ${\bar{C}}(\gamma ,\eta )=\frac{|\gamma |}{\eta }{\bar{C}}_{1}^{1-\frac{\gamma }{\eta }}$. Similar to the proof of Proposition 5.8, using the convexity of the map** $r >0 \mapsto r^{\frac{\gamma }{\eta }}$, we have

$$\begin{aligned} \int _{1}^{t}\left[ {\mathscr {D}}^{(\eta )}_{\varvec{\varepsilon }}(f(\tau ))\right] ^{\frac{\gamma }{\eta }}\mathrm {d}\tau&\geqslant (t-1)\left( \int _{1}^{t}{\mathscr {D}}_{\varvec{\varepsilon }}^{(\eta )}(f(\tau )) \frac{\mathrm {d}\tau }{t-1}\right) ^{\frac{\gamma }{\eta }}\\&=(t-1)^{1-\frac{\gamma }{\eta }}\left( \int _{1}^{t}{\mathscr {D}}_{ \varvec{\varepsilon }}^{(\eta )}(f(\tau ))\mathrm {d}\tau \right) ^{\frac{\gamma }{\eta }}, \qquad t \in (1,T). \end{aligned}$$

Therefore, in light of (5.12) and using (5.22), there exists $C_{\eta } >0$ depending only on $f_{\mathrm {in}}$ such that

$$\begin{aligned} \int _{1}^{t}\left[ {\mathscr {D}}^{(\eta )}_{\varvec{\varepsilon }}(f(\tau ))\right] ^{\frac{\gamma }{\eta }}\mathrm {d}\tau \geqslant C_{\eta }\left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{-\frac{\gamma }{\eta }}(t-1)^{1-\frac{\gamma }{\eta }} {\left( 1+t \right) ^{\frac{3\gamma }{\eta }}}, \qquad t \in (1,T)\,, \end{aligned}$$

where we used (1.14) and the fact that $\kappa _{0}(t) \geqslant 1-\varvec{\varepsilon }\chi ^{\star }$ for any $t \in [1,T)$. Inserting this into (5.23),

$$\begin{aligned} \varvec{y}(t)^{\frac{\gamma }{\eta }}-\varvec{y}(1)^{\frac{\gamma }{\eta }} \geqslant C_{\gamma ,\eta }\left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{1-2\frac{\gamma }{\eta }}(t-1)^{1-\frac{\gamma }{\eta }} {(1+t)^{3\frac{\gamma }{\eta }}}\,, \qquad t \in (1,T)\,, \end{aligned}$$

for some positive constant $C_{\gamma ,\eta } >0$ depending only on $\eta ,\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$. In other words

$$\begin{aligned} \varvec{y}(t) \leqslant \left( \varvec{y}(1)^{\frac{\gamma }{\eta }} + C_{\gamma ,\eta }\left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{1-2\frac{\gamma }{\eta }}(t-1)^{1-\frac{\gamma }{\eta }} {(1+t)^{3\frac{\gamma }{\eta }}}\right) ^{\frac{\eta }{\gamma }}, \end{aligned}$$

(5.24)

for any $t \in (1,T)$. In particular

$$\begin{aligned} \varvec{y}(t) \leqslant C_{\gamma ,\eta } \,\left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{\frac{\eta }{\gamma }-2}(t-1)^{\frac{\eta }{\gamma }-1}\, {(1+t)^{3}}\,, \qquad t \in {(2,T)}. \end{aligned}$$

We use this last estimate to sharpen the control of the third moment of f(t, v).

Lemma 5.9

For $\eta > 3+2|\gamma |$, one has

$$\begin{aligned} \sup _{t \in [2,T)}\varvec{m}_{3}(t) \leqslant C_{\eta ,\gamma }(f_{\mathrm {in}}) \left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{\frac{(\eta -3)(\eta -2\gamma )}{2\gamma \eta }} {+ \Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{3}}} . \end{aligned}$$

(5.25)

Proof

We use ideas similar to those introduced at the end of Proposition  5.8. For $\eta >3+2|\gamma |$, observe that

$$\begin{aligned} \begin{aligned} \Vert f(t) \Vert _{L^{1}_{3}}&\leqslant \Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{3}} + \Vert f(t) - {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{3}} \\&\leqslant \Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{3}} + \Vert f(t) - {\mathcal {M}}_{\varvec{\varepsilon }} \Vert ^{1-\frac{3}{\eta }}_{L^{1}}\Big (\Vert f(t) \Vert _{L^{1}_{\eta }} + \Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{\eta }}\Big )^{\frac{3}{\eta }}\\&\leqslant \Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{3}} + C_{\eta }\big ( 1+ t\big )^{\frac{3}{\eta }} \Vert f(t) - {\mathcal {M}}_{\varvec{\varepsilon }} \Vert ^{1-\frac{3}{\eta }}_{L^{1}}\\&\leqslant \Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{3}} + C_{\eta }\big ( 1+ t\big )^{\frac{3}{\eta }} {\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})^{\frac{1}{2}-\frac{3}{2\eta }} \,,\qquad t\geqslant 1\,, \end{aligned} \end{aligned}$$

where, in the last inequality, we used one side of the Csiszár-Kullback inequality (2.2). Let us note that $C_{\eta }$ does not depend on $\varvec{\varepsilon }$ since $\Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{\eta }}$ is uniformly bounded thanks to [2,  Lemma A.7]. Plugging into the aforementioned estimation for $\varvec{y}(t)={\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }})$, we obtain

$$\begin{aligned} \varvec{m}_{3}(t)\leqslant & {} \Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{3}} + C_{\eta }\left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{\frac{(\eta -3)(\eta -2\gamma )}{2\gamma \eta }}\times \\&\times (1+t)^{\frac{3}{\eta }}(t-1)^{\frac{(\eta -3)(\eta -\gamma )}{2\gamma \eta }}\, {(1+t)^{ \frac{3(\eta -3)}{2\eta }}}\,, \qquad t\in ({2},T)\,. \end{aligned}$$

Since $\eta > 3+2|\gamma |$, the function

$$\begin{aligned} t \geqslant 2 \longmapsto (1+t)^{\frac{3}{\eta }}(t-1)^{\frac{(\eta -3)(\eta -\gamma )}{2\gamma \eta }}\, {(1+t)^{ \frac{3(\eta -3)}{2\eta }}} \end{aligned}$$

is bounded by some positive constant $C_{\eta ,\gamma }$. We obtain then (5.25). $\square$

A simple consequence of the aforementioned Lemma is the following estimate on $\chi ^{\star }$.

Lemma 5.10

Assume that $\eta >3+2|\gamma |$, then there is a constant $C_{1}:=C_{1}(\gamma ,\eta ,f_{\mathrm{in}})$ independent of $\varvec{\varepsilon }$ and T such that

$$\begin{aligned} \chi ^{\star }(T)\left( 1-\varvec{\varepsilon }\chi ^{\star }(T)\right) ^{\alpha } \leqslant C_{1}, \qquad \qquad \alpha =\frac{(\eta -3)(\eta -2\gamma )}{2\eta (4+\gamma )} >0. \end{aligned}$$

(5.26)

Proof

Using Theorem 1.10 (with $s=3$) and the fact that $\sup _{\tau \in [0,2) }\varvec{m}_{3}(\tau ) \leqslant C(f_{\mathrm {in}})$ thanks to Proposition 3.11 (recall that (5.22) holds) we can use the previous estimate to conclude that

$$\begin{aligned} \chi ^{\star }=98\sup _{t \in [1,T)}\max \bigg (\Vert f(t)\Vert _{L^{\infty }}\,,\,\sup _{\varvec{\varepsilon }>0}\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{\infty }}\bigg ) \leqslant {{\tilde{C}}_{0}}\left( 1+\sup _{\tau \in [2,T)}\varvec{m}_{3}(\tau )\right) ^{-\frac{\gamma }{4+\gamma }}, \end{aligned}$$

which, with (5.25), gives

$$\begin{aligned} \chi ^{\star } \leqslant {C_{0}}\,\left( 1+\left( 1-\varvec{\varepsilon }\chi ^{\star } \right) ^{\frac{(\eta -3)(\eta -2\gamma )}{2\gamma \eta }}\right) ^{-\frac{\gamma }{4+\gamma }} \leqslant {C_{1}}\,\left( 1-\varvec{\varepsilon }\chi ^{\star }\right) ^{-\alpha } , \end{aligned}$$

where we used that $1-\varvec{\varepsilon }\chi ^{\star } \leqslant 1$. This gives (5.26). $\square$

We introduce the map**

$$\begin{aligned} \phi (x)=x\left( 1-\varvec{\varepsilon }\,x\right) ^{\alpha }, \qquad x \in (0, \varvec{\varepsilon }^{-1}). \end{aligned}$$

One has

$$\begin{aligned} \sup _{x\in (0, \varvec{\varepsilon }^{-1})}\phi (x)=\phi ({\bar{x}}) =\frac{\alpha ^{\alpha }}{\varvec{\varepsilon }(1+\alpha )^{1+\alpha }}, \qquad {\bar{x}}=\frac{1}{\varvec{\varepsilon }(1+\alpha )}. \end{aligned}$$

We define

$$\begin{aligned} \varvec{\varepsilon }_\star =\frac{\alpha ^\alpha }{2M(1+\alpha )^{\alpha +1}} \,, \end{aligned}$$

(5.27)

where $M >0$ is a (large) constant to be determined. We consider values $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star }]$ which ensure in particular that $M < \phi ({\bar{x}}).$ Now, in such a case, the equation $\phi (x)=M$ has two roots $x_{1}<{\bar{x}}<x_{2}$ in the interval $(0,\varvec{\varepsilon }^{-1})$. In particular, $\phi (x_{1})=M$ implies

$$\begin{aligned} x_{1}=\frac{M}{(1-\varvec{\varepsilon }x_{1})^{\alpha }}< \frac{M}{(1-\varvec{\varepsilon }{\bar{x}})^{\alpha }}=\left( 1+\frac{1}{\alpha }\right) ^{\alpha }M < {\frac{1}{2\varvec{\varepsilon }_{\star }}}. \end{aligned}$$

Therefore, the inequality $\phi (x) < M$ holds in the following two cases:

$$\begin{aligned} (i) \;\; \text { either }x \leqslant x_{1} < {\frac{1}{2\varvec{\varepsilon }_{\star }}}\,, \qquad \qquad (ii) \;\; \text { or }x \geqslant x_{2} > {\bar{x}} = \frac{1}{\varvec{\varepsilon }(1+\alpha )}\,. \end{aligned}$$

(5.28)

Let us now show that, choosing M large enough, the second case (ii) is an impossibility.

Lemma 5.11

Besides (5.27), assume that $M\geqslant C_{1}$ and

$$\begin{aligned} M\geqslant \frac{\chi ^{\star }(3)}{2} \left( \frac{\alpha }{1+\alpha }\right) ^{\alpha }. \end{aligned}$$

(5.29)

Then, for $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star })$, it holds that

$$\begin{aligned} \chi ^{\star }(T)\leqslant \chi ^{\star }(3) \leqslant x_{1} < {\frac{1}{2\varvec{\varepsilon }_{\star }}}, \qquad T\in (2,3]. \end{aligned}$$

Proof

Notice that (5.29) means that $\varvec{\varepsilon }_{\star } \leqslant \frac{1}{\chi ^\star (3)(1+\alpha )}.$ Applying Theorem 1.10 with $s=3$ on the interval [1, 3), one has

$$\begin{aligned} \sup _{t \in [1,3)}\left\| f(t)\right\| _{L^{\infty }} \leqslant C\,\bigg (1 + \sup _{t \in [1,3)}\varvec{m}_{3}(t)\bigg )^{-\frac{\gamma }{4+\gamma }} \end{aligned}$$

for some positive C depending on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$ and this last quantity is finite and independent of $\varvec{\varepsilon }$ thanks to Proposition 3.11 since $\varvec{m}_{3}(0) <\infty .$ Therefore,

$$\begin{aligned} \chi ^\star (3)=98\sup _{t \in [1,3)}\max \bigg (\Vert f(t)\Vert _{L^{\infty }}\,,\,\sup _{\varvec{\varepsilon }>0}\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{\infty }}\bigg ) < \infty \,, \end{aligned}$$

depends only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$. Under the additional constraint (5.29), we observe that for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star }]$ it holds ${\bar{x}}\geqslant \chi ^{\star }(3)$, which excludes the case (ii). By the aforementioned binary option, one gets the desired conclusion. $\square$

This argument shows the existence of a trap** region which can be extended to any $T>3$.

Lemma 5.12

Assume (5.27) and (5.29) are in force. Then, defining

$$\begin{aligned} T^{\star } := \sup \Big \{ T>2 \; \big | \chi ^{\star }(T)\leqslant x_{1}\Big \}\,, \end{aligned}$$

one can choose M large enough (explicit) such that $T^{\star }=\infty$ for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star })$.

Proof

We already saw in Lemma 5.11 that $T^{\star }\geqslant 3$. We argue by contradiction considering that $T^{\star }<\infty$. In all the sequel, we will denote by u(t) a function of $t \geqslant 0$ which is such that $\lim _{t\rightarrow 0^{+}}u(t)=0$ (i.e. $u(t)=o(1)$) and that may change from line to line. Recalling and integrating the moment inequality (3.10) (with $\delta =1$ and $s=3$) in the time interval $(T^\star ,T^\star +t )$, it follows that

since the latter three terms in the right-hand side can be made as small as desired when $t \rightarrow 0$. In other words,

$$\begin{aligned} \sup _{\tau \in [2,T^\star +t )}\varvec{m}_{3}(\tau ) = \sup _{\tau \in [2,T^\star )}\varvec{m}_{3}(\tau ) + {u(t)}\,,\qquad t \in (0,1]\,. \end{aligned}$$

(5.30)

Using Theorem 1.10 applied on the interval $[1,T^\star + t )$, the fact that $\sup _{\tau \in [0,2) }\varvec{m}_{3}(\tau ) \leqslant C(f_{\mathrm {in}})\,,$ and the continuous growth of the third moment (5.30), one is led to

$$\begin{aligned} \chi ^\star (T^\star +t )&\leqslant C\,\left( 1 + \sup _{\tau \in [2,T^\star + t )}\varvec{m}_{3}(\tau )\right) ^{-\frac{\gamma }{4+\gamma }}\\&= C\,\left( 1 + \sup _{\tau \in [2,T^\star )}\varvec{m}_{3}(\tau )\right) ^{-\frac{\gamma }{4+\gamma }} + {u(t)}\,, \end{aligned}$$

for some positive C depending on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$ and $H(f_{\mathrm{in}})$. Consequently, one can use (5.25) with $T=T^\star$ to get

$$\begin{aligned}&\left( 1 + \sup _{\tau \in [2,T^\star )}\varvec{m}_{3}(\tau )\right) ^{-\frac{\gamma }{4+\gamma }} \\&\quad \leqslant \left( 1+C_{\eta ,\gamma }(f_{\mathrm {in}}) \left( 1-\varvec{\varepsilon }\chi ^{\star }(T^{\star })\right) ^{\frac{(\eta -3)(\eta -2\gamma )}{2\gamma \eta }}+ \Vert {\mathcal {M}}_{\varvec{\varepsilon }} \Vert _{L^{1}_{3}}\right) ^{-\frac{\gamma }{4+\gamma }}\\&\quad \leqslant 2^{-\frac{\gamma }{4+\gamma }-1}C_{\eta ,\gamma }^{-\frac{\gamma }{4+\gamma }} \left( 1-\varvec{\varepsilon }\chi ^{\star }(T^{\star })\right) ^{\frac{(\eta -3)(\eta -2\gamma )}{2\gamma \eta }\frac{-\gamma }{4+\gamma }}\\&\quad +2^{-\frac{\gamma }{4+\gamma }-1} \left( 1+\Vert {\mathcal {M}}_{\varvec{\varepsilon }}\Vert _{L^{1}_{3}}\right) ^{-\frac{\gamma }{4+\gamma }}. \end{aligned}$$

We deduce from this that there is some ${\overline{C}} >0$ independent of $\varvec{\varepsilon }$ and M such that

$$\begin{aligned}&\chi ^{\star }(T^{\star }+t ) \leqslant {\overline{C}}\left( 1+\big (1-\varvec{\varepsilon }\chi ^{\star }(T^{\star })\big )^{-\alpha }\right) +u(t)\nonumber \\&\leqslant 2{\overline{C}}\big (1-\varvec{\varepsilon }\chi ^{\star }(T^{\star })\big )^{-\alpha } + {u(t)}\,, \end{aligned}$$

(5.31)

where we used again that $1-\varvec{\varepsilon }\chi ^{\star }(T^{\star }) \leqslant 1.$ Notice that, by definition of $T^{\star }$,

$$\begin{aligned} \big (1-\varvec{\varepsilon }\chi ^{\star }(T^{\star })\big )^{-\alpha } {\leqslant } \big (1-\varvec{\varepsilon }x_{1}\big )^{-\alpha }\,, \end{aligned}$$

where ${1-\varvec{\varepsilon }\chi ^{\star }(T^{\star }) \geqslant } 1-\varvec{\varepsilon }x_{1} \geqslant 1-\varvec{\varepsilon }_{\star }x_{1}>\tfrac{1}{2}$. Thus, $\left( 1-\varvec{\varepsilon }\chi ^{\star }(T^{\star })\right) ^{-\alpha } \in [1,2^{\alpha }]$ and, for t small enough, (5.31) implies that

$$\begin{aligned} \chi ^{\star }(T^{\star }+t ) \leqslant 3{\overline{C}}\big (1-\varvec{\varepsilon }\chi ^{\star }(T^{\star })\big )^{-\alpha } \leqslant 3{\overline{C}}\left( 1-\varvec{\varepsilon }x_{1}\right) ^{-\alpha }. \end{aligned}$$

(5.32)

Now set

$$\begin{aligned} M:=\max \left( \frac{\chi ^{\star }(3)}{2}\left( \frac{\alpha }{1+\alpha } \right) ^{\alpha },3{\overline{C}} , C_{1} \right) . \end{aligned}$$

One deduces from (5.32) that

$$\begin{aligned} \chi ^{\star }(T^{\star }+t ) \leqslant M(1-\varvec{\varepsilon }x_{1})^{-\alpha }=x_{1}\,, \end{aligned}$$

which is a contradiction since, by definition of $T^{\star }$, $\chi ^{\star }(T^{\star }+t ) > x_{1}.$ Thus, for the above choice of M, we must have that $T^{\star }=\infty$.$\square$

We have all in hands to conclude.

Proof of Theorem 1.7

The previous Lemma exactly means that, for some explicit $\varvec{\varepsilon }_{\star } >0$ (associated to the above choice of M), one has

$$\begin{aligned} \chi ^{\star }(T) \leqslant x_{1} < {\frac{1}{2\varvec{\varepsilon }_{\star }}}\,, \qquad \qquad \forall \;T > 2\,, \qquad \forall \varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star }). \end{aligned}$$

This proves in particular that

$$\begin{aligned} \sup _{t\geqslant 1}\Vert f(t)\Vert _{L^{\infty }} \leqslant {\frac{1}{196\varvec{\varepsilon }_{\star }}}, \end{aligned}$$

which is independent of $\varvec{\varepsilon }$. This gives the no saturation property

$$\begin{aligned} \kappa _{0} = 1 - \varvec{\varepsilon }\,\sup _{t\geqslant 1}\Vert f(t) \Vert _{\infty }>0, \qquad \forall \,\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star }]. \end{aligned}$$

At this stage, we can resume the proof of Proposition 5.8 to get the desired result. $\square$

6 Upgrading the Convergence

We explain in this section how the rate of convergence can be upgraded to a stretched exponential rate whenever the initial datum satisfies a more stringent decay in terms of large velocities decay. As before, our strategy is based upon suitable interpolations. Notations are those of Section 5.1 and we follow at first the interpolation procedure developed in [29, Section 5]. Namely, we begin by improving the interpolation inequality between ${\mathscr {D}}_{\varvec{\varepsilon }}^{(\gamma )}$ and ${\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}$ provided by inequality (5.5).

Lemma 6.1

For a given $a>0$ and $q >0$ define, for any $0 \leqslant g \leqslant \varvec{\varepsilon }^{-1}$,

$$\begin{aligned} \Gamma ^{a,q}_{\varvec{\varepsilon }}(g)=\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}|v-v_{*}|^{2} \exp (a|v-v_{*}|^{q})\varvec{\** }_{\varvec{\varepsilon }}[g](v,v_{*})\mathrm {d}v\mathrm {d}v_{*}\,, \end{aligned}$$

where $\varvec{\** }_{\varvec{\varepsilon }}$ is defined by formula (2.5). Then for any $\gamma <0$ one has that

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(\gamma )}(g) \geqslant \frac{1}{2}\left[ \frac{1}{a} \log \left( \frac{\Gamma _{\varvec{\varepsilon }}^{(a,q)}(g)}{{\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(g)} \right) \right] ^{\frac{\gamma }{q}}\,{\mathscr {D}}^{(0)} _{\varvec{\varepsilon }}(g), \end{aligned}$$

(6.1)

where ${\mathscr {D}}_{\varvec{\varepsilon }}^{(\gamma )}(g)$ is defined by formula (5.1).

Proof

For a given $R>0$, we set ${\mathcal {Z}}_{a,R}{=}\left\{ (v,v_{*}) \in {\mathbb {R}}^{3}\times {\mathbb {R}}^{3};|v-v_{*}| \leqslant \left( \frac{R}{a}\right) ^{\frac{1}{q}}\right\}$ and denote by ${\mathcal {Z}}_{a,R}^{c}$ its complementary in ${\mathbb {R}}^{6}.$ We see that

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(g)= & {} \frac{1}{2}\int _{{\mathcal {Z}}_{a,R}}|v-v_{*}|^{\gamma }\,|v-v_{*}|^{-\gamma } \,|v-v_{*}|^{2}\varvec{\** }_{\varvec{\varepsilon }}[g](v,v_{*})\,\mathrm {d}v\mathrm {d}v_{*}\\&+\frac{1}{2}\int _{{\mathcal {Z}}_{a,R}^{c}}\exp \left( -a|v-v_{*}|^{q} \right) \exp \left( a|v-v_{*}|^{q}\right) \,|v-v_{*}|^{2}\varvec{\** }_{\varvec{\varepsilon }}[g](v,v_{*})\,\mathrm {d}v\mathrm {d}v_{*}\\\leqslant & {} \left( \frac{R}{a}\right) ^{\frac{|\gamma |}{q}}{\mathscr {D}}_{\varvec{\varepsilon }}^{(\gamma )}(g) +\frac{1}{2}\exp (-R)\Gamma _{\varvec{\varepsilon }}^{(a,q)}(g)\,. \end{aligned}$$

We also notice that for any $a,q>0$, we have that $1\leqslant \exp \left( a|v-v_{*}|^q\right)$, and therefore $2{\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(g) \leqslant \Gamma ^{a,q}_{\varvec{\varepsilon }}(g)$. Thus, the choice

$$\begin{aligned} R:= \log \left( \frac{\Gamma _{\varvec{\varepsilon }}^{a,q}(g)}{{\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(g)}\right) \geqslant \log 2 >0 \end{aligned}$$

is possible, and yields

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(g) \leqslant \left[ \log \left( \frac{\Gamma _{\varvec{\varepsilon }}^{(a,q)}(g)}{{\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(g)}\right) \right] ^{\frac{|\gamma |}{q}}\,a^{\frac{\gamma }{q}}\,{{\mathscr {D}}_{\varvec{\varepsilon }}^{(\gamma )}(g)} +\frac{1}{2}{\mathscr {D}}^{(0)}_{\varvec{\varepsilon }}(g)\,, \end{aligned}$$

which completes the proof. $\square$

Remark 6.2

Applying this inequality to a weak solution f(t, v) to (1.11.10) and assuming that the initial datum $f_{\mathrm {in}}$ and $\varvec{\varepsilon }>0$ are such that (5.19) holds, together with the estimate

$$\begin{aligned} \sup _{t\geqslant 1}\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(t)) \leqslant {\bar{\Gamma }}\,, \end{aligned}$$

(6.2)

we expect that the relative entropy

$$\begin{aligned} \varvec{y}(t)={\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}) \end{aligned}$$

satisfies a differential inequality of the type

$$\begin{aligned} \dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{y}(t) \leqslant -\frac{{\bar{\lambda }}_{0}}{2}\left[ \frac{1}{a} \log \left( \frac{{\bar{\Gamma }}}{{\bar{\lambda }}_{0}\varvec{y}(t)}\right) \right] ^{\frac{\gamma }{q}}\,\varvec{y}(t), \qquad t \geqslant 1, \end{aligned}$$

leading to an estimate like

$$\begin{aligned} \varvec{y}(t) \leqslant A\exp (-Bt^{\frac{q}{q-\gamma }})\,, \qquad t \geqslant 1, \end{aligned}$$

for some positive constant $A,B >0$. We will see that, even if we cannot prove directly (6.2), the (at most) slowly increasing growth of $\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(t))$ will be such that the above decay still holds.

Following the paths of Section 5, we first look for suitable upper bound for $\Gamma _{\varvec{\varepsilon }}^{(a,q)}(g).$ We proceed as in Lemma 5.3 to get the following result.

Lemma 6.3

For any $0 \leqslant g \leqslant \varvec{\varepsilon }^{-1}$ satisfying (5.3) and any $a>0$, $q \in (0,1)$ one has

$$\begin{aligned} \Gamma _{\varvec{\varepsilon }}^{(a,q)}(g) \leqslant \frac{{32}}{\kappa _{0}(g)}\left\| g\,\mu _{a,q}\right\| _{L^{1}_{2}}\,\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{2}\left| \nabla \sqrt{g(v)}\right| ^{2}\mu _{a,q}(v)\,\mathrm {d}v, \end{aligned}$$

(6.3)

where

$$\begin{aligned} \mu _{a,q}(v)=\exp \left( a\, \langle v\rangle ^{q}\right) , \qquad v \in {\mathbb {R}}^{3}. \end{aligned}$$

(6.4)

Proof

Recalling definition (2.5), we see that

$$\begin{aligned} \Gamma _{\varvec{\varepsilon }}^{(a,q)}(g)=\int _{{\mathbb {R}}^{6}}|v-v_{*}|^{2}\exp (a|v-v_{*}|^{q}) \,F\,F_{*}\left| \Pi (v-v_{*})\left[ \nabla h-\nabla h_{*}\right] \right| ^{2}\,\mathrm {d}v\mathrm {d}v_{*}, \end{aligned}$$

where $h(v)=\log (g(v))-\log (1-\varvec{\varepsilon }g(v))$ and $F=g(1-\varvec{\varepsilon }g)$. Using the obvious estimate

$$\begin{aligned} \left| \Pi (v-v_{*})\left[ \nabla h-\nabla h_{*}\right] \right| ^{2} \leqslant 2|\nabla h|^{2}+2|\nabla h_{*}|^{2}, \end{aligned}$$

and $|v-v_{*}|^{2}\exp (a|v-v_{*}|^{q}) \leqslant 2\langle v\rangle ^{2}\mu _{a,q}(v)\langle v_{*}\rangle ^{2}\mu _{a,q}(v_{*})$ since $q \in (0,1)$, one deduces that

$$\begin{aligned} \begin{aligned} \Gamma _{\varvec{\varepsilon }}^{(a,q)}(g)&\leqslant {8} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{2}\mu _{a,q}(v)F(v)\,|\nabla h(v)|^{2}\,\mathrm {d}v\int _{{\mathbb {R}}^{3}} F(v_*)\,\mu _{a,q}(v_{*})\langle v_{*}\rangle ^{2}\mathrm {d}v_{*}\\&\leqslant {8}\int _{{\mathbb {R}}^{3}}\frac{|\nabla g(v)|^{2}}{g(1-\varvec{\varepsilon }g)}\langle v\rangle ^{2}\mu _{a,q}(v)\,\mathrm {d}v\int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{2}g_{*}\mu _{a,q}(v_{*})\mathrm {d}v_{*}. \end{aligned} \end{aligned}$$

This yields the result. $\square$

As for Proposition 5.5, on the basis of (6.3) and (6.1), it is useful to get a uniform in time upper bound of the above Fisher information with exponential weights along solutions to (1.11.10). Before doing so, let us introduce the following objects:

Definition 6.4

Given $a,q >0$, we recall definition (6.4). Then, for any nonnegative measure function $g\,:\,{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}$, we set

$$\begin{aligned} \Upsilon _{a,q}(g) :=\int _{{\mathbb {R}}^{3}}g^{2}(v)\mu _{a,q}(v)\,\mathrm {d}v, \qquad \vartheta _{a,q}(g) :=\int _{{\mathbb {R}}^{3}}g(v)\mu _{a,v}(v)\,\mathrm {d}v. \end{aligned}$$

Given $s \geqslant 0$, we also introduce

$$\begin{aligned} {\overline{\Upsilon }}_{a,q,s}(g) :=\int _{{\mathbb {R}}^{3}}g^{2}(v)\langle v\rangle ^{s}\mu _{a,q}(v)\,\mathrm {d}v, \qquad {\overline{\vartheta }}_{a,q,s}(g) :=\int _{{\mathbb {R}}^{3}}g(v)\langle v\rangle ^{s}\mu _{a,q}(v)\,\mathrm {d}v. \end{aligned}$$

If f(t, v) is a weak-solution to (1.11.10), we will moreover simply write

$$\begin{aligned} \Upsilon _{a,q}(t) :=\Upsilon _{a,q}(f(t)), \qquad \vartheta _{a,q}(t) :=\vartheta _{a,q}(f(t)), \qquad t \geqslant 0\,, \end{aligned}$$

with similar notations for ${\overline{\Upsilon }}_{a,q,s}(t),{\overline{\vartheta }}_{a,q,s}(t)$. We also set

$$\begin{aligned} \varvec{\Pi }_{a,q}(t) :=\tfrac{1}{2}\Upsilon _{a,q}(t)+\vartheta _{a,q}(t). \end{aligned}$$

Estimates on the evolution of the above family of moments are easily deduced from Theorem 1.9 since we keep track, for the evolution of $\varvec{E}_{s}(t)$, of the dependency with respect to s. Namely, one has the following proposition, with a proof quite similar to that of [22, Corollary 4.1].

Proposition 6.5

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Let $a >0$ and $0< q <\frac{4+2\gamma }{8-\gamma }$. Assume that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\exp \left( {2^{q(1+\frac{1}{|\gamma |})}}a\langle v\rangle ^{q}\right) f_{\mathrm{in}}(v)\,\mathrm {d}v= {\vartheta _{{\tilde{a}},q}(f_{\mathrm{in}})} < \infty \,, \qquad \qquad {{\tilde{a}}=2^{\frac{q(1+|\gamma |)}{|\gamma |}}a}. \end{aligned}$$

Then there exists $\varvec{C}_{a,q} >0$ depending only on a, q and $f_{\mathrm{in}}$ such that

$$\begin{aligned} \varvec{\Pi }_{a,q}(t) \leqslant \varvec{C}_{a,q}\left( t^{-\frac{3}{2}}+t\right) \,, \qquad t >0\,. \end{aligned}$$

Proof

As in [22, Corollary 4.1], we notice that

$$\begin{aligned} \varvec{\Pi }_{a,q}(t)=\sum _{n=0}^{\infty }\frac{a^{n}}{n!}\varvec{E}_{nq}(t)\,, \end{aligned}$$

so that thanks to Theorem 1.9

$$\begin{aligned} \varvec{\Pi }_{a,q}(t) \leqslant \Big ( t^{-\frac{3}{2}}+t \Big )\sum _{n=0}^{\infty }\frac{a^{n}}{n!}\varvec{C}_{nq}\,. \end{aligned}$$

Consequently, in order to prove the result, we just need to show that the sum is finite. Using now (1.15), there is $\beta _{1} >0$ such that

$$\begin{aligned} \varvec{C}_{nq} \leqslant \beta _{1}\left[ \left( \beta _{1}\,q\,n\right) ^{{{\frac{8-\gamma }{4+2\gamma }}(nq+\gamma -2)+1}} +2^{\frac{nq}{|\gamma |}} \, {(1 + nq)^{\frac{5}{2}}} \, \varvec{m}_{nq}(0)\right] \,\quad (nq >6+|\gamma |). \end{aligned}$$

Clearly, for n large enough $(1+nq)^{\frac{5}{2}} \leqslant c_{0}2^{nq}$ for some universal $c_{0} >0$, so that

$$\begin{aligned} \varvec{C}_{nq} \leqslant 2\beta _{1} \left( \beta _{1}\,q\,n\right) ^{nb+\ell } + \beta _{1}c_{0}2^{nq(1+\frac{1}{|\gamma |})}\,\varvec{m}_{nq}(0)\,, \end{aligned}$$

with $b=\frac{8-\gamma }{4+2\gamma }q$ and $\ell = \frac{8-\gamma }{4+2\gamma } (\gamma -2)+1.$ Using Stirling formula and d’Alembert’s ratio test, one sees easily that, if $b <1$, then the sum

$$\begin{aligned} \sum _{n=0}^{\infty }\tfrac{a^{n}}{n!} \left( \beta _{1}\,q\,n\right) ^{nb+\ell } \quad \text { is finite for any } a >0\,, \end{aligned}$$

whereas

$$\begin{aligned} \sum _{n=0}^{\infty }\tfrac{a^{n}}{n!} 2^{\frac{nq(1+|\gamma |)}{|\gamma |}}\varvec{m}_{nq}(0)=\int _{{\mathbb {R}}^{3}}\exp \left( {2^{q(1+\frac{1}{|\gamma |})}}a\langle v\rangle ^{q}\right) f_{\mathrm{in}}(v)\,\mathrm {d}v < \infty \,. \end{aligned}$$

This proves the result.$\square$

Remark 6.6

From the above proof, one sees that, if $q= {\frac{4+2\gamma }{8-\gamma }}$, then the above result still holds if $2^{\frac{q}{4}}a\beta _{1}q e <1$.

We need in the sequel an analogue of Lemma 5.4.

Lemma 6.7

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Then, given $a,q >0$ and any $s >\frac{3}{2}$, there exists $C_{s}(f_{\mathrm {in}}) >0$ depending on s, $f_{\mathrm{in}}$, (but not on a, q) such that, for any $k \geqslant 0$ and any $t\geqslant 0$,

$$\begin{aligned}&-\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }[f(t)]\,f(t,v)\big (1+|\log f(t,v)|\big )\mu _{a,q}(v)\,\mathrm {d}v \nonumber \\&\quad \leqslant C_{s}(f_{\mathrm {in}})\bigg (\sqrt{ {\overline{\vartheta }}_{2a,q,2(k+s)}(t)}+{\overline{\Upsilon }}_{a,q,k}(t)\nonumber \\&\qquad +\left( {\overline{\vartheta }}_{\frac{3}{2}a,q,\frac{3}{2}k}(t) +{\overline{\Upsilon }}_{\frac{3}{2}a,q,\frac{3}{2}k}(t) \right) ^{\frac{2}{3}}{\Big (1+\frac{1}{t}\Big )}\bigg )\,, \end{aligned}$$

(6.5)

and

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k+\gamma }f(t,v)\left( 1+\left| \log f(t,v)\right| \right) \mu _{a,q}(v)\,\mathrm {d}v \nonumber \\&\quad \leqslant C_{s}(f_{\mathrm {in}})\left( \sqrt{ {\overline{\vartheta }}_{2a,q,2(k+s+\gamma )}(t)}+{\overline{\Upsilon }}_{a,q,k+\gamma }(t)\right) \,. \end{aligned}$$

(6.6)

Proof

The proof is very similar to that of Lemma 5.4 and is based upon (5.10). We use the same notations as in Lemma 5.4 and use the splitting $\varvec{c}_{\gamma }[f]=\varvec{c}_{\gamma }^{(1)}[f]+\varvec{c}_{\gamma }^{(2)}[f]$. One has

$$\begin{aligned}&-\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }[f]\,f\left( 1+|\log f|\right) \mu _{a,{q} }(v)\,\mathrm {d}v \\&\quad \leqslant -C_{\frac{4}{3},\frac{3}{2}}\int _{{\mathbb {R}}^{3}} \langle v\rangle ^{k}\varvec{c}_{\gamma }^{(1)}[f]\left( f^{\frac{2}{3}} +f^{\frac{4}{3}}\right) \mu _{a,q}(v)\,\mathrm {d}v \\&\quad -C_{2,2}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }^{(2)}[f]\left( \sqrt{f}+f^{2}\right) \mu _{a,q}(v)\,\mathrm {d}v\,. \end{aligned}$$

As in Lemma 5.4, a simple use of Cauchy-Schwarz inequality yields, for any $s > \frac{3}{2}$,

$$\begin{aligned}&-\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{k}\varvec{c}_{\gamma }^{(2)}[f(t)]\left( \sqrt{f}+f^{2}\right) \mu _{a,q}(v)\mathrm {d}v\\&\leqslant C_{s}(f_{\mathrm {in}})\left( \sqrt{ {\overline{\vartheta }}_{2a,q,2(k+s)}(t)}+{\overline{\Upsilon }}_{a,q,k}(t)\right) , \end{aligned}$$

for some positive constant depending only on $s, \Vert f_{\mathrm {in}}\Vert _{L^1_{2}}$. In the same way, as in Lemma 5.4, we deduce from Hölder’s inequality, and Proposition 3.15 that

This proves (6.5). Now, the proof of (6.6) follows the same lines as that of (5.9). $\square$

Proposition 6.8

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Let $b,q >0$ be given, with $q < {\frac{4+2\gamma }{8-\gamma }}$. Assume moreover that

$$\begin{aligned} f_{\mathrm {in}} \in L^{1}\left( {\mathbb {R}}^{3},\mu _{2{{\tilde{b}}}+\delta ,q}(v)\,\mathrm {d}v\right) ,\qquad {\tilde{b}} := { 2^{\frac{q(1+|\gamma |)}{|\gamma |}}b}\,, \end{aligned}$$

for some $\delta >0$. Then, for any $t_{0} >0$, there exists $C >0$ depending on $b,\delta ,q,t_{0}$ and $f_{\mathrm{in}}$, such that

$$\begin{aligned} \int _{t_{0}}^{t}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{\gamma }\,\mu _{b,q}(v)\left| \nabla \sqrt{f(\tau ,v)}\right| ^{2}\,\mathrm {d}v \leqslant C\big ( 1+t \big )^{2}\,, \qquad t>t_{0} >0. \end{aligned}$$

Proof

Let us fix $b,q >0$. We investigate the evolution of

$$\begin{aligned} \varvec{S}_{b,q}(t):=\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)f(t,v)\log f(t,v)\,\mathrm {d}v \end{aligned}$$

for a solution $f=f(t,v)$ to (1.11.10). To simplify notations, we write $F=F(t,v)=f(1-\varvec{\varepsilon }f)$. One checks, similar to (5.13), that

$$\begin{aligned} \dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{S}_{b,q}(t)= & {} \dfrac{\mathrm {d}}{\mathrm {d}t}\vartheta _{b,q}(t)+\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\nabla \cdot \left( \varvec{\Sigma }[f]\nabla f\right) \log f\,\mathrm {d}v\\&-\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\nabla \cdot \left( \varvec{b}[f]F\right) \log f\,\mathrm {d}v , \end{aligned}$$

with

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\nabla \cdot \left( \varvec{\Sigma }[f]\nabla f\right) \log f\,\mathrm {d}v= & {} -\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\varvec{\Sigma }[f]\nabla f\cdot \frac{\nabla f}{f}\,\mathrm {d}v\\&+\int _{{\mathbb {R}}^{3}}\nabla \cdot \big (\varvec{\Sigma }[f]\nabla \mu _{b,q}\big )\,\big [f\log f-f\big ]\,\mathrm {d}v\,, \end{aligned}$$

and

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\nabla \cdot \left( \varvec{b}[f]F\right) \log f\,\mathrm {d}v= & {} -bq \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{q-2}\,F\log f\,\big (\varvec{b}[f]\cdot v\big )\mu _{b,q}(v)\, \,\mathrm {d}v\\&+b\,q \int _{{\mathbb {R}}^{3}}\big (f-\frac{\varvec{\varepsilon }}{2}f^{2}\big )\langle v\rangle ^{q-2}\big (\varvec{b}[f]\cdot v\big )\,\mu _{b,q}(v)\,\mathrm {d}v\\&{+}\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\left( f-\frac{\varvec{\varepsilon }}{2}f^{2}\right) \varvec{c}_{\gamma }[f]\,\mathrm {d}v. \end{aligned}$$

For the latter, we notice that

$$\begin{aligned} \nabla \mu _{b,q}(v)&=b\,q\,v\langle v\rangle ^{q-2}\mu _{b,q}(v)\,,\qquad \text {and}\\ \nabla \cdot \big (\varvec{\Sigma }[f]\nabla \mu _{b,q}\big )&=b\,q\,\mu _{b,q}(v)\Big (\langle v\rangle ^{q-2}\varvec{B}[f]\cdot v +\langle v\rangle ^{q-4}\mathrm {Trace}\big (\varvec{\Sigma }[f]\cdot \varvec{A}_{\mu }(v)\big )\Big ), \end{aligned}$$

with $\varvec{A}_{\mu }(v)=\langle v\rangle ^{2}\mathbf {Id}+\left[ (q-2)+b\,q\,\langle v\rangle ^{q}\right] \,v\otimes v$.

As in the proof of Proposition 5.5, using that both $|\varvec{b}[f]\cdot v|$ and $\frac{1}{2}|\varvec{B}[f]\cdot v|$ satisfy (3.44), and using now that

$$\begin{aligned} \mathrm {Trace}\left( \varvec{\Sigma }[f]\cdot \varvec{A}_{\mu }(v)\right) \leqslant {C_{b,q}} \langle v\rangle ^{q+4+\gamma }\Vert f\Vert _{L^{1}_{\gamma +2}}\,, \end{aligned}$$

one deduces the following analogue of (5.165.18),

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t}\varvec{S}_{b,q}(t)-\dfrac{\mathrm {d}}{\mathrm {d}t}\vartheta _{b,q}(t) +K_{0}\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\langle v\rangle ^{\gamma }\frac{|\nabla f|^{2}}{f}\,\mathrm {d}v \nonumber \\&\quad \leqslant C_{b,q}(f_{\mathrm {in}})\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{2q+\gamma }f\left( 1+\left| \log f\right| \right) \mu _{b,q}(v)\,\mathrm {d}v\nonumber \\&\qquad -C_{b,q}(f_{\mathrm {in}})\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{q}\varvec{c}_{\gamma }[f]\,f\left( 1+\left| \log f\right| \right) \mu _{b,q}(v)\,\mathrm {d}v , \end{aligned}$$

(6.7)

for some positive constant $C_{b,q}(f_{\mathrm {in}})$ depending on b, q and $f_{\mathrm {in}}$ only through $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$. We use now the results of Lemma 6.7 to get for $s=2$ that

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t}\varvec{S}_{b,q}(t)-\dfrac{\mathrm {d}}{\mathrm {d}t}\vartheta _{b,q}(t) +K_{0}\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\langle v\rangle ^{\gamma }\frac{|\nabla f|^{2}}{f}\,\mathrm {d}v \nonumber \\&\quad \leqslant C_{b,q} (f_{\mathrm {in}})\left( \sqrt{ {\overline{\vartheta }}_{2b,q,2r+4}(t)}+{\overline{\Upsilon }}_{b,q,r}(t) +\left( {\overline{\vartheta }}_{\frac{3}{2}b,q,\frac{3}{2}q}(t) +{\overline{\Upsilon }}_{\frac{3}{2}b,q,\frac{3}{2}q}(t)\right) ^{\frac{2}{3}}{\Big (1+\tfrac{1}{t}\Big )}\right) , \end{aligned}$$

(6.8)

where $r=\max (2q+\gamma ,q)$. For any $\delta >0$ and $t_{0} >0$, since ${\overline{\vartheta }}_{2b,q,2r+4}(t) \leqslant C_{\delta }\,{\vartheta }_{2b+\delta ,q}(t)$ and similarly for ${\overline{\Upsilon }}_{b,q,r}(t)$ and the remainder terms, one has that, for $t\geqslant t_0$,

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t}\varvec{S}_{b,q}(t)-\dfrac{\mathrm {d}}{\mathrm {d}t}\vartheta _{b,q}(t) +K_{0}\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\langle v\rangle ^{\gamma }\frac{|\nabla f|^{2}}{f}\,\mathrm {d}v \nonumber \\&\quad \leqslant C_{b,q,\delta }(f_{\mathrm {in}},t_{0})\left( \sqrt{\vartheta _{2b+\delta ,q}(t)}+ {\Upsilon }_{b+\delta ,q}(t)+\left( \vartheta _{\frac{3}{2}b+\delta ,q}(t)+ {\Upsilon }_{\frac{3}{2}b+\delta ,q}(t)\right) ^{\frac{2}{3}}\right) \,. \end{aligned}$$

(6.9)

Using now Proposition 6.5, assuming that $q < \frac{4+2\gamma }{8-\gamma }$ and ${\vartheta }_{2{\tilde{b}}+\delta ,q}(0) < \infty$ (after renaming $\delta >0$) we deduce that for $t\geqslant t_0$ and $\delta >0$,

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\varvec{S}_{b,q}(t)-\dfrac{\mathrm {d}}{\mathrm {d}t}\vartheta _{b,q}(t) +K_{0}\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)\langle v\rangle ^{\gamma }\frac{|\nabla f|^{2}}{f}\,\mathrm {d}v \leqslant C_{\delta ,b,q,t_{0}}(f_{\mathrm {in}})\big (1+t\big )\,, \end{aligned}$$

for some positive constant $C_{\delta ,b,q,t_{0}}(f_{\mathrm {in}})$ depending only on $\delta ,b,q,t_{0}$ and $f_{\mathrm {in}}$. Integrating this inequality over $(t_{0},t)$ yields

$$\begin{aligned}&K_{0}\int _{t_{0}}^{t}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }\frac{|\nabla {f(\tau ,v)}|^{2}}{f(\tau ,v)}\mu _{b,q}(v)\,\mathrm {d}v \\&\quad \leqslant \varvec{S}_{b,q}({t_0})-\varvec{S}_{b,q}(t)+\vartheta _{b,q}(t) + \frac{1}{2}C_{\delta ,b,q,t_{0}}(f_{\mathrm {in}})\big (1+t \big )^{2}. \end{aligned}$$

Arguing as in the proof of [27,  Eq. (B.3), Lemma B.4], introducing $A=\{v \in {\mathbb {R}}^{3}\;;\;f(t,v) <1\}$, one checks easily that

$$\begin{aligned} -\varvec{S}_{b,q}(t)= & {} -\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)f(t,v)\left| \log f(t,v)\right| \,\mathrm {d}v \\&+2\int _{A}\mu _{b,q}(v)f(t,v)\log \left( \frac{1}{f(t,v)}\right) \,\mathrm {d}v. \end{aligned}$$

Introducing then $B:=\{v \in {\mathbb {R}}^{3}\;;\;f(t,v)\geqslant \exp \left( -{3}\,b\langle v\rangle ^{q}\right) \}$, one splits the integral over A into the integral over $A \cap B$ and $A \cap B^{c}$. On the one hand, for $v \in A \cap B$, $\log \frac{1}{f(t,v)} \leqslant {3}\,b\langle v\rangle ^{q}$ and, for any $\delta >0$, there exists $C_{\delta }=C(\delta ,q,b) >0$ such that

$$\begin{aligned} 2\int _{A \cap B}\mu _{b,q}(v)f(t,v)\log \left( \frac{1}{f(t,v)}\right) \,\mathrm {d}v\leqslant & {} C_{\delta }\int _{{\mathbb {R}}^{3}}\mu _{b+\delta ,q}(v)f(t,v)\,\mathrm {d}v\\= & {} C_{\delta }\vartheta _{b+\delta ,q}(t). \end{aligned}$$

On the other hand, for $v \in A\cap B^{c}$, using that $x\log \frac{1}{x} \leqslant \frac{2}{e}\sqrt{x}$, one has

$$\begin{aligned} f(t,v)\log \left( \frac{1}{f(t,v)}\right) \leqslant \frac{2}{e}\sqrt{f(t,v)} \leqslant \frac{2}{e}\exp \left( {-\frac{3}{2}}b\langle v\rangle ^{q}\right) \,, \end{aligned}$$

so that

$$\begin{aligned}&\int _{A\cap B^{c}}\mu _{b,q}(v)f(t,v)\log \left( \frac{1}{f(t,v)}\right) \,\mathrm {d}v\\&\leqslant \frac{2}{e}\int _{{\mathbb {R}}^{3}}\exp \left( - {\frac{1}{2}}b\langle v\rangle ^{q}\right) \,\mathrm {d}v=:C_{b,q} <\infty . \end{aligned}$$

This shows that, for any $\delta >0$,

$$\begin{aligned} -\varvec{S}_{b,q}(t) \leqslant -\int _{{\mathbb {R}}^{3}}\mu _{b,q}(v)f(t,v)|\log f(t,v)|\,\mathrm {d}v+C_{\delta }\vartheta _{b+\delta ,q}(t)+2C_{b,q}\,. \end{aligned}$$

Moreover, we deduce from Eq. (6.6) in Lemma 6.7 together with Proposition 6.5 that $\varvec{S}_{b,q}( {t_0})$ is finite under our assumption on $f_{\mathrm{in}}$ and

$$\begin{aligned}&K_{0}\int _{ {t_0}}^{t}\mathrm {d}\tau \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\gamma }\mu _{b,q}(v)|\nabla \sqrt{f(\tau ,v)}|^{2}\,\mathrm {d}v \leqslant C\big ( 1+t \big )^{2}, \end{aligned}$$

for some finite $C >0$ depending on $b,q,\delta , {t_0} >0$ and $f_{\mathrm {in}}$. $\square$

We deduce from this the following estimate for $\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(t))$.

Corollary 6.9

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Let $a >0$ and $0<q <{\frac{4+2\gamma }{8-\gamma }}$, and assume that for some $\delta >0$,

$$\begin{aligned} \int _{{\mathbb {R}}^{3}} f_{\mathrm {in}}(v) \exp \left( \left( 2a+\delta \right) { 2^{\frac{q(1+|\gamma |)}{|\gamma |}}}\langle v\rangle ^{q}\right) \,\mathrm {d}v < \infty . \end{aligned}$$

Then, there exists $C_{\delta ,a,q}(f_{\mathrm {in}})>0$ depending only on $\delta ,a,q$ and $f_{\mathrm {in}}$ such that

$$\begin{aligned} \int _{t_{0}}^{t}\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(\tau ))\mathrm {d}\tau \leqslant C_{\delta ,a,q}(f_{\mathrm {in}})\,\sup _{t_0\leqslant \tau \leqslant t} \frac{\vartheta _{a+\delta ,q}(\tau )}{\kappa _{0}(\tau )}\; (1+t)^{2} \,,\qquad 0< t_{0} < t\,, \end{aligned}$$

where we recall that $\kappa _{0}(\tau )=1-\varvec{\varepsilon }\,\Vert f(\tau )\Vert _{L^{\infty }}$, $\tau \geqslant 0.$

Proof

The proof follows from Lemma 6.3, Proposition 6.8 with $b=a$, and the fact that $\langle v\rangle ^{2}\mu _{a,q}(v) \leqslant {C_{\delta ,q}} {\langle v\rangle ^{\gamma }}\mu _{a+\delta ,q}(v)$ for any $\delta >0$. $\square$

Theorem 6.10

Assume that $-2< \gamma < 0$ and let a nonnegative initial datum $f_{\mathrm {in}}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$ be given. For $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$, let $f(t,\cdot )$ be a weak-solution to (1.11.10). Let $a_0 >0$ and $0<q <{\frac{4+2\gamma }{8-\gamma }}$, and assume that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}f_{\mathrm {in}}(v) \exp \big (a_0\langle v\rangle ^{q}\big )\,\mathrm {d}v < \infty . \end{aligned}$$

Then, there exists some explicit $\lambda >0$ depending only on $a_0,q$ and $f_{\mathrm{in}}$ such that, for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star })$ (where $\varvec{\varepsilon }_{\star }$ is prescribed by Theorem 1.7),

$$\begin{aligned} {\mathcal {H}}_{\varvec{\varepsilon }}(f(t))|{\mathcal {M}}_{{\vec {\varvec{\varepsilon }}}}) \leqslant \max \left( 1,{\mathcal {H}}_{{\vec {\varvec{\varepsilon }}}}(f_{\mathrm {in}}|{\mathcal {M}}_{{\vec {\varvec{\varepsilon }}}}) \right) \exp \left( -\lambda \,t^{\frac{q}{q-\gamma }}\right) , \qquad t \geqslant 2. \end{aligned}$$

As a consequence,

$$\begin{aligned} \left\| f(t)-{\mathcal {M}}_{\varvec{\varepsilon }}\right\| _{L^{1}} \leqslant \sqrt{\max \left( 2,2{\mathcal {H}}_{\varvec{\varepsilon }}(f_{\mathrm {in}}|{\mathcal {M}}_{\varvec{\varepsilon }}) \right) }\,\exp \left( -\frac{\lambda }{2}\,t^{\frac{q}{q-\gamma }}\right) \,, \qquad t \geqslant 2\,. \end{aligned}$$

Proof

We first observe that, thanks to Theorem 1.7 and under the assumptions on the initial datum $f_{\mathrm {in}}$, there is $\varvec{\varepsilon }_{\star } \in (0,\varvec{\varepsilon }_{0}]$ depending only on $f_{\mathrm {in}}$ such that for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star }]$,

$$\begin{aligned} \kappa _{0} = 1 - \varvec{\varepsilon }\,\sup _{t\geqslant 1}\Vert f(t) \Vert _{\infty }>0. \end{aligned}$$

Let us write

$$\begin{aligned} \varvec{y}(t)={\mathcal {H}}_{\varvec{\varepsilon }}(f(t)|{\mathcal {M}}_{\varvec{\varepsilon }}), \qquad t\geqslant 0. \end{aligned}$$

One uses then (6.1) and (5.19) which, by Proposition 5.2 and Theorem 1.7, actually holds for $-2<\gamma <0$. We first deduce that, for $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{\star })$, $t \geqslant 1$, and $a >0$, $q \in (0,1)$,

$$\begin{aligned} {\mathscr {D}}_{\varvec{\varepsilon }}^{(\gamma )}(f(t))&\geqslant \frac{1}{2}\left[ \frac{1}{a} \log \left( \frac{\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(t))}{{\mathscr {D}}_{\varvec{\varepsilon }}^{(0)} (f(t))}\right) \right] ^{\frac{\gamma }{q}}\,{\mathscr {D}}^{(0)} _{\varvec{\varepsilon }}(f(t))\\&\geqslant \frac{{\bar{\lambda }}_{0}}{2}\left[ \frac{1}{a} \log \left( \frac{\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(t))}{{\bar{\lambda }}_{0}\varvec{y} (t)}\right) \right] ^{\frac{\gamma }{q}}\,\varvec{y}(t), \end{aligned}$$

where we recall that we already know that $\frac{\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(t))}{{\mathscr {D}}_{\varvec{\varepsilon }}^{(0)}(f(t))} >1$, so that $\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(t)) \geqslant {{\bar{\lambda }}_{0}\, \varvec{y}(t)}$. We deduce then that

$$\begin{aligned} \dfrac{\mathrm {d}}{\mathrm {d}t}\varvec{y}(t) \leqslant -\frac{{\bar{\lambda }}_{0}}{2}\left[ \frac{1}{a} \log \left( \frac{\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(t))}{{\bar{\lambda }}_{0} \varvec{y}(t)}\right) \right] ^{\frac{\gamma }{q}}\,\varvec{y}(t), \qquad t \geqslant 1. \end{aligned}$$

Using Grönwall’s Lemma, we get

$$\begin{aligned} \varvec{y}(t) \leqslant \varvec{y}(1) \exp \left\{ -\frac{{\bar{\lambda }}_{0}}{2a^{\frac{\gamma }{q}}}\int _{1}^{t}\left[ - \log \left( \frac{{\bar{\lambda }}_{0}\varvec{y}(\tau )}{\Gamma _{\varvec{\varepsilon }}^{(a, q)}(f(\tau ))}\right) \right] ^{\frac{\gamma }{q}}\mathrm {d}\tau \right\} , \qquad t\geqslant 1.\nonumber \\ \end{aligned}$$

(6.10)

We introduce

$$\begin{aligned} I_{q}(t):=\int _{1}^{t}\left[ - \log \left( \frac{{\bar{\lambda }}_{0}\varvec{y}(\tau )}{\Gamma _{\varvec{\varepsilon }}^{(a, q)}(f(\tau ))}\right) \right] ^{\frac{\gamma }{q}}\mathrm {d}\tau , \qquad t \geqslant 2\,, \end{aligned}$$

so that

$$\begin{aligned} I_{q}(t)&\geqslant \int _{\frac{t}{2}}^{t}\left[ - \log \left( \frac{{ \bar{\lambda }}_{0}\varvec{y}(\tau )}{\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(\tau ))} \right) \right] ^{\frac{\gamma }{q}}\mathrm {d}\tau \\&=\int _{\frac{t}{2}}^{t}\left[ \log \Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(\tau )) -\log {\bar{\lambda }}_{0}-\log \varvec{y}(\tau )\right] ^{\frac{\gamma }{q}} \mathrm {d}\tau , \qquad t \geqslant 2. \end{aligned}$$

Assume now that there is $t_{0} >2$ and some $m {>} 0$ such that

$$\begin{aligned} \varvec{y}(t_{0}) \geqslant \exp \left( -\left( \frac{t_{0}}{2}\right) ^{m}\right) \,. \end{aligned}$$

(6.11)

Then, since $\tau \mapsto \varvec{y}(\tau )$ is non increasing, one has

$$\begin{aligned} \varvec{y}(\tau ) \geqslant \varvec{y}(t_{0}) \geqslant \exp \left( -\tau ^{m}\right) \geqslant \exp \left( -t_{0}^{m}\right) \,, \qquad \tau \in \left( \frac{t_{0}}{2},t_{0}\right) \,, \end{aligned}$$

and

$$\begin{aligned} I_{q}(t_{0}) \geqslant \int _{\frac{t_{0}}{2}}^{t_{0}}\left[ \log \Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(\tau ))-\log {\bar{\lambda }}_{0} +t_{0}^{m}\right] ^{\frac{\gamma }{q}} \mathrm {d}\tau . \end{aligned}$$

Using now that, for any $\alpha \in {\mathbb {R}},$ the function $r >\exp (-\alpha ) \mapsto (\alpha +\log r)^{\frac{\gamma }{q}}$ is convex, and applying it with $\alpha =-\log {\bar{\lambda }}_{0}+t_{0}^{m}$, we deduce from Jensen’s inequality that

$$\begin{aligned} I_{q}(t_{0})&\geqslant \frac{t_{0}}{2}\int _{\frac{t_{0}}{2}}^{t_{0}} \left[ \log \Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(\tau ))+\alpha \right] ^{\frac{\gamma }{q}} \frac{2\,\mathrm {d}\tau }{t_{0}}\\&\geqslant \frac{t_{0}}{2}\left[ t_{0}^{m}-\log {\bar{\lambda }}_{0}+ \log \left( \frac{2}{t_{0}}\int _{\frac{t_{0}}{2}}^{t_{0}} \Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(\tau ))\,\mathrm {d}\tau \right) \right] ^{\frac{\gamma }{q}}. \end{aligned}$$

Using Corollary 6.9 together with Proposition 6.5, choosing parameters $a,\delta >0$ such that $a_0=\Big (2a+\delta \Big ){ 2^{\frac{q(1+|\gamma |)}{|\gamma |}}}$, there are positive constants $C_{0},C_{1}>0$ (independent of $t_{0}$) such that

$$\begin{aligned} \int _{\frac{t_{0}}{2}}^{t_{0}}\Gamma _{\varvec{\varepsilon }}^{(a,q)}(f(\tau ))\mathrm {d}\tau \leqslant {C_{0}\left( 1+t_{0}\right) ^{{3}}}\,, \end{aligned}$$

so that

$$\begin{aligned} I_{q}(t_{0}) \geqslant \frac{t_{0}}{2}\Big [t_{0}^{m}-\log {\bar{\lambda }}_{0}+\log (2C_{0})+{3}\log (1+t_{0}) - \log t_0\Big ]^{\frac{\gamma }{q}}\,. \end{aligned}$$

Consequently, there exists $C_{2} >0$ such that

$$\begin{aligned} I_{q}(t_{0}) \geqslant C_{2}\, t_{0}^{1+\frac{m\gamma }{q}}. \end{aligned}$$

Inserting this in (6.10), we now deduce that

$$\begin{aligned} \varvec{y}(t_{0}) \leqslant \varvec{y}(1)\exp \left( -\frac{{\bar{\lambda }}_{0}\,C_{2}}{2a^{\frac{\gamma }{q}}}\, t_{0}^{1+m\frac{\gamma }{q}}\right) . \end{aligned}$$

(6.12)

Since we proved that assumption (6.11) implies estimate (6.12), we see that for any $t >2$ and any $m {>} 0$, we have the following alternative:

• (i) either $\;\;\varvec{y}(t) \leqslant \exp \left( -\left( \frac{t}{2}\right) ^{m}\right)$,

• (ii) or $\;\;\varvec{y}(t) \leqslant \varvec{y}(1)\exp \left( -\frac{{\bar{\lambda }}_{0}C_{2}}{2a^{\frac{\gamma }{q}}}\,t^{1+m\frac{\gamma }{q}}\right)$.

At this state, choosing $m >0$ so that $m=1+m\frac{\gamma }{q}$ (that is $m=\frac{q}{q-\gamma }$), we see that

$$\begin{aligned} \varvec{y}(t) \leqslant \max \big (1,\varvec{y}(1)\big )\exp \big (-c_{a}t^{m}\big ), \qquad t \geqslant 2\,, \end{aligned}$$

for some explicit $c_{a}:=\min \left( 2^{-m},\frac{{\bar{\lambda }}_{0}C_{2}}{2a^{\frac{\gamma }{q}}}\right) .$ This concludes the proof. $\square$

Appendices

Appendix A. Regularity estimates

We collect here several a priori regularity estimates for the solutions to (1.11.10). Clearly, it is possible to interpolate between $L^{1}$ and $L^{\infty }$ thanks to Theorem 1.10 to obtain a control on the $L^{p}$-norms with $1<p<\infty$. We adopt another approach here which consists in directly investigating the evolution of the $L^{p}$-norms:

Proposition A.1

Assume that $-2<\gamma <0$. Let $p\geqslant 1$ and $f_{\mathrm {in}}\in L^p({\mathbb {R}}^{3}) \cap L^{2}({\mathbb {R}}^{3})\cap L^1_{\varvec{z}_{p}}({\mathbb {R}}^{3})$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$, with

$$\begin{aligned} \varvec{z}_{p}:=\frac{1}{p}{\left\{ \begin{array}{ll} (3p-2)|\gamma | &{}\quad \text { if } \,\gamma \leqslant -1\,,\\ 3p-1+\gamma &{}\quad \text { if }\, \gamma \in (-1,0)\,.\end{array}\right. } \end{aligned}$$

(A.1)

Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and let f(t, v) be a weak solution to (1.11.10). Then, there exists some constant $C_p(f_{\mathrm {in}})$ depending on p and $f_{\mathrm {in}}$ such that, for every $T>0$,

$$\begin{aligned} \sup _{t \in [0,T)}\left\| f(t,\cdot )\right\| ^p_{L^{p}} + \int _{0}^{T}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{\frac{\gamma }{2}}\left| \nabla f^{\frac{p}{2}}(t,v)\right| ^{2} \,\mathrm {d}v \mathrm {d}t \leqslant C_{p}(f_{\mathrm {in}})\left( 1+T\right) ^{p+1}.\nonumber \\ \end{aligned}$$

(A.2)

Proof

We start with the formulation (1.11.10)

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t} f = \nabla \cdot \big (\,\varvec{\Sigma }[f]\, \nabla f - \varvec{b}[f]\, f(1-\varvec{\varepsilon }f)\big ),\\&f(t=0)= f_{\mathrm {in}}\,. \end{aligned}\right. \end{aligned}$$

(A.3)

For $p \geqslant 1$, multiplying this identity with $f^{p-1}(t,v)$ and integrating over ${\mathbb {R}}^{3}$, one deduces

$$\begin{aligned}&\frac{1}{p}\dfrac{\mathrm {d}}{\mathrm {d}t}\int _{{\mathbb {R}}^{3}} f^{p}(t,v)\,\mathrm {d}v + (p-1) \int _{{\mathbb {R}}^{3}} f^{p-2}(t,v) \, [ \varvec{\Sigma }[f(t)](v) \nabla f(t,v) ] \cdot \nabla f(t,v) \,\mathrm {d}v \\&\quad =(p-1)\int _{{\mathbb {R}}^{3}} f^{p-2}(t,v) F(t,v) \,\varvec{b}[f(t)](v) \cdot \nabla f(t,v), \qquad F=f(1-\varvec{\varepsilon }f). \end{aligned}$$

Using the coercivity estimate in Proposition 2.3, $|F| \leqslant f$, and noticing that $f^{\frac{p}{2}}\nabla f^{\frac{p}{2}}=\frac{p}{2}f^{p-1}\nabla f$, we obtain, after integration between 0 and T,

$$\begin{aligned}&\frac{1}{p}\int _{{\mathbb {R}}^{3}} f^{p}(T,v)\, \,\mathrm {d}v + (p-1) \, K_0 \int _0^T \mathrm {d}t \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{\gamma } f^{p-2}(t,v)\, |\nabla f(t,v)|^2\,\mathrm {d}v \\&\quad \leqslant \frac{1}{p}\int _{{\mathbb {R}}^{3}} f^{p}_{\mathrm {in}}(v)\, \,\mathrm {d}v + \frac{2(p-1)}{p} \int _0^T\mathrm {d}t \int _{{\mathbb {R}}^{3}} f^{\frac{p}{2}}(t,v) \, \left| \varvec{b}[f](t,v)\right| \, |\nabla f^{\frac{p}{2}}(t,v)|\,\mathrm {d}v\,. \end{aligned}$$

Note that from this point on, the estimates do not use $\varvec{\varepsilon }$ and are thus similar to that of the usual Landau equation. Since $f^{p-2}|\nabla f|^{2}=\frac{4}{p^{2}}|\nabla f^{\frac{p}{2}}|^{2}$, using Young’s inequality, we get

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}} f^{p}(T,v)\,\mathrm {d}v + \frac{2K_{0}\,(p-1)}{p}\int _0^T\mathrm {d}t \int _{{\mathbb {R}}^{3}} \langle v \rangle ^{\gamma } |\nabla f^{\frac{p}{2}}(t,v)|^2\,\mathrm {d}v \\&\quad \leqslant \int _{{\mathbb {R}}^{3}} f^{p}_{\mathrm {in}}(v)\,\mathrm {d}v + \frac{p(p-1)}{2 K_0} \int _0^T \mathrm {d}t \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{-\gamma } f^{p}(t,v)\,\left| \varvec{b}[f](t,v)\right| ^{2}\,\mathrm {d}v\\&\quad \leqslant \int _{{\mathbb {R}}^{3}} f^{p}_{\mathrm {in}}(v)\,\mathrm {d}v +\frac{p(p-1)C(f_{\mathrm {in}})^{2}}{2K_{0}}\int _{0}^{T}\mathrm {d}t\int _{{\mathbb {R}}^{3}}f^{p}(t,v)\langle v\rangle ^{\max (-\gamma ,2+\gamma )}\,\mathrm {d}v , \end{aligned}$$

where we used (3.42), (3.43) and (3.38) in the last term, $C(f_{\mathrm {in}})$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{2}}$ and $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$. Since

$$\begin{aligned} \langle v\rangle ^{\gamma } \left| \nabla f^{\frac{p}{2}}(t,v)\right| ^2 \geqslant \frac{1}{2}\left| \nabla \left( \langle v\rangle ^{\gamma /2} \, f^{\frac{p}{2}}(t,v)\right) \right| ^2-\left| \nabla (\langle v\rangle ^{\frac{\gamma }{2}})\right| ^2 f^{p}(t,v)\,, \end{aligned}$$

we get that

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}} f^{p}(T,v)\, \,\mathrm {d}v + \frac{p-1}{p} \, K_0 \int _0^T \mathrm {d}t\int _{{\mathbb {R}}^{3}}\left| \nabla \left( \langle v\rangle ^{\gamma /2} \, f^{\frac{p}{2}}(t,v)\right) \right| ^2 \,\mathrm {d}v \nonumber \\&\quad \leqslant \int _{{\mathbb {R}}^{3}} f^{p}_{\mathrm {in}}(v)\,\mathrm {d}v + C_{0}p(p-1) \int _0^T \mathrm {d}t \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{\max (-\gamma ,2+\gamma )} \, f^{p}(t,v)\,\mathrm {d}v , \end{aligned}$$

(A.4)

for some positive constant $C_{0}$ depending only on $\Vert f_{\mathrm {in}}\Vert _{L^{1}_{2}}$, $\Vert f_{\mathrm{in}}\Vert _{L^{2}}$ and $H(f_{\mathrm{in}})$.

Choosing $q >1$, $a >0$ and $\theta \in (0,1)$ such that $q'\,\theta =1$ and

$$\begin{aligned} q(p - \theta )= 3p,\qquad \frac{1}{q}+\frac{1}{q'}=1, \end{aligned}$$

and applying Hölder’s inequality, we see that, with $\varpi =\max (-\gamma ,2+\gamma )$,

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^{3}} \langle v\rangle ^{\max (-\gamma ,2+\gamma )} \, f^{p}(t,v)\,\mathrm {d}v \\&\quad = \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{\varpi + a} \, f(t,v)^{\theta } \, \langle v\rangle ^{- a} \, f(t,v)^{p - \theta }\,\mathrm {d}v\\&\quad \leqslant \bigg [ \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{(\varpi + a) q'} \, f(t,v)^{q' \theta }\,\mathrm {d}v \bigg ]^{\frac{1}{q'}} \, \bigg [ \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{- a q} \, f(t,v)^{q(p- \theta )}\,\mathrm {d}v \bigg ]^{\frac{1}{q}} \\&\quad \leqslant \bigg [ \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{\frac{\varpi + a}{\theta } } \, f(t,v)\,\mathrm {d}v \bigg ]^{\theta } \, \bigg [ \int _{{\mathbb {R}}^{3}} \bigg (\langle v\rangle ^{- \frac{a q}{6}} \, f(t,v)^{\frac{p}{2}} \bigg )^6 \,\mathrm {d}v\bigg ]^{\frac{1}{q}}\,. \end{aligned} \end{aligned}$$

Using the Sobolev inequality (2.14), we conclude that

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}} \langle v\rangle ^{\max (-\gamma ,2 +\gamma )} \,f^p(t,v)\,\mathrm {d}v\\&\leqslant C_{\mathrm {Sob}}^{\frac{6}{q}}\varvec{m}_{\frac{\varpi +a}{\theta }}(t)^{\theta } \, \, \bigg [ \int _{{\mathbb {R}}^{3}}\bigg | \nabla \bigg (\langle v\rangle ^{-\frac{a q}{6}} \, f^{\frac{p}{2}}(t,v) \bigg ) \bigg |^2\mathrm {d}v \bigg ]^{3 \, (1 - \theta )} . \end{aligned}$$

At this point, observe that $\theta = \frac{2p}{3p-1}$ and select a in such a way that $aq = 3|\gamma |$. Thus,

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}} \langle v\rangle ^{\max (-\gamma ,2 +\gamma )} \,f^p(t,v)\, \,\mathrm {d}v\nonumber \\&\leqslant C_{p}\varvec{m}_{\varvec{z}_{p}}(t)^{\frac{2p}{3p-1}} \bigg [ \int _{{\mathbb {R}}^{3}} \bigg | \nabla \bigg (\langle v\rangle ^{\frac{\gamma }{2}} \, f^{\frac{p}{2}}(t,v) \bigg ) \bigg |^2\,\mathrm {d}v \bigg ]^{ \frac{3p-3}{3p-1}}, \end{aligned}$$

(A.5)

with $\varvec{z}_{p}=\frac{\varpi +a}{\theta }$. Note that, for $\gamma \in (-2,0)$, $\varvec{z}_{p} \leqslant \frac{6p-4}{p} < 6$. Then, using Young’s inequality it holds for any $\delta >0$,

$$\begin{aligned} x^{\frac{2p}{3p-1}} \, y^{\frac{3p-3}{3p-1}} \leqslant \delta ^{-\frac{p-1}{2}}\, x^{p} + \delta ^{\frac{1}{3}} \,y\,, \qquad \forall \, x,\,y >0. \end{aligned}$$

Choosing $C_{0}\,C_{p}\,\delta ^{\frac{1}{3}} = \frac{1}{2p^{2}} \, K_0$, we get after combining (A.4) and (A.5) that

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}} f^{p}(T,v)\,\mathrm {d}v + \frac{p-1}{2p} \, K_0 \int _0^T \mathrm {d}t\int _{{\mathbb {R}}^{3}}\left| \nabla \left( \langle v\rangle ^{\frac{\gamma }{2}} \, f^{\frac{p}{2}}(t,v)\right) \right| ^2 \,\mathrm {d}v \\&\quad \leqslant \int _{{\mathbb {R}}^{3}} f^{p}_{\mathrm {in}}(v)\,\mathrm {d}v + {\tilde{C}}_{p}\int _{0}^{T}\varvec{m}_{\varvec{z}_{p}}(t)^{p}\mathrm {d}t. \end{aligned}$$

One sees from the second part of (1.14) in Theorem 1.9 that

$$\begin{aligned} \int _{0}^{T}\varvec{m}_{\varvec{z}_{p}}(t)^{p}\mathrm {d}t \leqslant C_{p}(f_{\mathrm{in}})\left( 1+T\right) ^{p+1}\,, \end{aligned}$$

for some $C_{p}(f_{\mathrm{in}})$ depending only on $\Vert f_{\mathrm{in}}\Vert _{L^{1}_{2}}$, and $H(f_{\mathrm{in}})$ and p. We deduce from this estimate (A.2). $\square$

For simplicity of notations, we introduce here $L^{1}_{\infty }({\mathbb {R}}^{3}) :=\bigcap _{s\geqslant 0}L^{1}_{s}({\mathbb {R}}^{3})$ as the space of integrable functions with finite moments of any order and, more generally,

$$\begin{aligned} W^{1,p}_{\infty }({\mathbb {R}}^{3})=\bigcap _{s\geqslant 0}W^{1,p}_{s}({\mathbb {R}}^{3}) \end{aligned}$$

for any $p\geqslant 1$.

Corollary A.2

Assume $-2<\gamma <0$ and let $q_0 >1$. We assume that $f_{\mathrm {in}}\in L^{q_0}({\mathbb {R}}^3) \cap L^{2}({\mathbb {R}}^{3})\cap L^1_{\infty }({\mathbb {R}}^{3})$ satisfies (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and f(t, v) be a weak solution to (1.11.10). Then, for any $m >0, q \in [1,q_{0})$ and for any $T>0$, there exists some constant C depending on $q_0$, m, q, $f_{\mathrm{in}}$ such that

$$\begin{aligned} \sup _{t \in [0, T]} \int _{{\mathbb {R}}^{3}} \langle v\rangle ^{m} \, f(t,v)^q\, \,\mathrm {d}v < C\, (1 + T)^{1 + q_0}\,. \end{aligned}$$

(A.6)

Proof

This is a direct consequence of an interpolation between the spaces $L^{q_0}$ and $L^1_{s}$, using Proposition A.1 and the universal growth of the $L^{1}$-moments in Theorem 1.9. $\square$

This corollary will be used in a crucial way to prove the following proposition. We introduce here the notation $L^{\infty -0}({\mathbb {R}}^{3}) :=\bigcap _{q \geqslant 1}L^{q}({\mathbb {R}}^{3})$.

Proposition A.3

Assume $-2<\gamma <0$ and let $f_{\mathrm {in}}\in L^1_{\infty }({\mathbb {R}}^{3}) \bigcap L^{\infty -0}({\mathbb {R}}^3)$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and f(t, v) be a weak solution to (1.11.10). For any choice of $m \geqslant 0$ and $p \geqslant 2$, if $f_{\mathrm{in}} \in W^{1,p}_{m}({\mathbb {R}}^{3})$, then there is some $C_T(f_{\mathrm {in}})$ depending on m, p, T and $f_{\mathrm {in}}$ such that

$$\begin{aligned}&\sup _{t\in [0,T]} \int _{{\mathbb {R}}^3} \langle v\rangle ^{m} \, | \nabla f(t,v) |^p \,\mathrm {d}v+ \sum _{i=1}^3 \int _0^T\mathrm {d}t\int _{{\mathbb {R}}^3} \langle v\rangle ^{ m +\gamma } \, |\partial _i f|^{p-2}\, |\nabla \partial _i f|^2 \,\mathrm {d}v \nonumber \\&\quad \leqslant C_T(f_{\mathrm {in}}). \end{aligned}$$

(A.7)

Proof

Taking the derivative of (1.11.10) with respect to a component $i \in \{1,2,3\}$, we end up with the equation

$$\begin{aligned} \left\{ \begin{aligned} \partial _{t} (\partial _i f)&= \nabla \cdot \big (\,\varvec{\Sigma }[f]\, \nabla (\partial _i f) + [\partial _i \varvec{\Sigma }[f] ] \nabla f - \varvec{b}[f]\, \partial _i F - [\partial _i \varvec{b}[f]] \, F \big ),\\ \partial _i f(0, \cdot )&=\partial _i f_{\mathrm {in}}\,. \end{aligned}\right. \end{aligned}$$

(A.8)

Compute

$$\begin{aligned}&\frac{1}{p} \frac{\mathrm {d}}{\mathrm {d}t} \sum _i \int _{{\mathbb {R}}^3} \langle v\rangle ^m\, |\partial _i f|^{p} \,\mathrm {d}v \\&\quad = (p-1)\sum _i \int _{{\mathbb {R}}^3} \langle v\rangle ^m\,|\partial _i f|^{p-2}\bigg \{ \nabla (\partial _i f) \cdot \varvec{b}[f] \partial _i F \\&\qquad +\nabla (\partial _{i}f) \cdot \partial _{i}\varvec{b}[f]\,F - \varvec{\Sigma }[f] \nabla (\partial _i f) \nabla (\partial _i f) - \nabla (\partial _i f) \partial _i \varvec{\Sigma }[f] \nabla f\bigg \} \,\mathrm {d}v\\&\qquad - \sum _i \int _{{\mathbb {R}}^3} |\partial _i f|^{p-2} \partial _i f \, \varvec{\Sigma }[f] \nabla (\partial _i f) \nabla \langle v\rangle ^m \,\mathrm {d}v\\&\qquad - \sum _i \int _{{\mathbb {R}}^3} \nabla \langle v\rangle ^m\, |\partial _i f|^{p-2} \partial _i f \, [\partial _i \varvec{\Sigma }[f]] \nabla f \,\mathrm {d}v\\&\qquad + \sum _i \int _{{\mathbb {R}}^3} \nabla \langle v\rangle ^m \cdot \varvec{b}[f]\, \, |\partial _i f|^{p-2} \partial _i f \, \partial _i F \,\mathrm {d}v\\&\qquad + \sum _i \int _{{\mathbb {R}}^3} \nabla \langle v\rangle ^m \cdot [\partial _i \varvec{b}[f]] \, |\partial _i f|^{p-2} \partial _i f \, F \,\mathrm {d}v. \end{aligned}$$

Using the coercivity estimate in Proposition 2.3 and Young’s inequality we see that

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t} \sum _i \int _{{\mathbb {R}}^3} \langle v\rangle ^m {|\partial _i f|^{p}} \,\mathrm {d}v + \frac{K_0}{2}\;{p(p-1)} \sum _i \int _{{\mathbb {R}}^3} \langle v\rangle ^{ m +\gamma } \, |\partial _i f|^{p-2}\, |\nabla \partial _i f|^2 \,\mathrm {d}v\nonumber \\&\quad \leqslant C_p \int _{{\mathbb {R}}^3} \langle v\rangle ^{m + |\gamma |} |\nabla f|^{p-2} \bigg [ |\nabla f|^2 |\nabla \varvec{\Sigma }[f] |^2 + F^2\, |\nabla \varvec{b}[f]|^2\nonumber \\&\qquad + |\nabla F|^2\, | \varvec{b}[f]|^2 + \langle v\rangle ^{-2} |\nabla f|^2 | \varvec{\Sigma }[f] |^2 \bigg ] \,\mathrm {d}v\nonumber \\&\qquad + C_p \int _{{\mathbb {R}}^3} \langle v\rangle ^{m-1} \, |\nabla f|^{p} \, |\nabla \varvec{\Sigma }[f]| \,\mathrm {d}v\nonumber \\&\qquad + C_p \int _{{\mathbb {R}}^3} \langle v\rangle ^{m-1}\, |\nabla f|^{p-1} \, \bigg ( |\varvec{b}[f]|\, |\nabla F| + |\nabla \varvec{b}[f]| \, F \bigg )\,\mathrm {d}v, \end{aligned}$$

(A.9)

where $C_p>0$ only depends on p and m. Then, we observe that

$$\begin{aligned}&\big |\varvec{\Sigma }[f] (v)\big | \leqslant C(f_{\mathrm {in}})\langle v\rangle ^{2 +\gamma }\,,\nonumber \\&|\nabla F(v)| \leqslant C \, (1+f(v))\, |\nabla f(v)|, \end{aligned}$$

(A.10)

$$\begin{aligned}&\big | \nabla \varvec{\Sigma }[f] (v) \big | + \big | \varvec{b}[f](v) \big | \leqslant C(f_{\mathrm {in}})\, \langle v\rangle ^{ \max \{ 0,1 +\gamma \} }\,, \nonumber \\&|\nabla \varvec{b}[f](t,v)| \leqslant C\, (|\cdot |^{- |\gamma |} * f)(v), \end{aligned}$$

(A.11)

and (A.9) leads to

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t} \sum _i \int _{{\mathbb {R}}^3} \langle v\rangle ^m {|\partial _i f|^{p}} \,\mathrm {d}v + \frac{K_0}{2} \; {p(p-1)} \sum _i \int _{{\mathbb {R}}^3} \langle v\rangle ^{ m +\gamma } \, |\partial _i f|^{p-2}\, |\nabla \partial _i f|^2 \,\mathrm {d}v \nonumber \\&\quad \leqslant C_p(f_{\mathrm {in}}) \bigg ( \int _{{\mathbb {R}}^3} \langle v\rangle ^{ m + \max (|\gamma |, 2 - |\gamma |)} (1+f)^2 \, |\nabla f|^{p} \,\mathrm {d}v\nonumber \\&\qquad + \int _{{\mathbb {R}}^3} \langle v\rangle ^{m +|\gamma |} f^2 \, |\nabla f|^{p-2} \, {( |\cdot |^{- |\gamma |} * f)^{2}}\, \,\mathrm {d}v\nonumber \\&\qquad + \int _{{\mathbb {R}}^3} \langle v\rangle ^{ m -1} f \, |\nabla f|^{p-1} \, ( |\cdot |^{- |\gamma |} * f)\, \,\mathrm {d}v\bigg ). \end{aligned}$$

(A.12)

For $p >2$, using Young’s inequality $x\,y \leqslant x^a + y^{\frac{a}{a-1}}$ with $a=\frac{p}{p-2}$ for the second term of the right-hand side of the aforementioned estimate, and with $a =\frac{p}{p-1}$ for the third term, we conclude that for some $r_1, r_2 \geqslant 0$

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} \sum _i \int _{{\mathbb {R}}^3} \langle v\rangle ^m {|\partial _i f|^{p}} \,\mathrm {d}v + \frac{K_0}{2} {p(p-1)} \sum _i \int _{{\mathbb {R}}^3} \langle v\rangle ^{ m +\gamma } \, |\partial _i f|^{p-2}\, |\nabla \partial _i f|^2 \,\mathrm {d}v \nonumber \\&\quad \leqslant C_p(f_{\mathrm {in}}) \bigg ( \int _{{\mathbb {R}}^3} \langle v\rangle ^{ m + \max (|\gamma |, 2 - |\gamma |)} (1+f)^2 \, |\nabla f|^{p} \,\mathrm {d}v\nonumber \\&\quad + \int _{{\mathbb {R}}^3} \langle v\rangle ^{r_1} f^2 \, ( |\cdot |^{- |\gamma |} * f)^p\, \,\mathrm {d}v+ \int _{{\mathbb {R}}^3} \langle v\rangle ^{r_2} f \, {( |\cdot |^{- |\gamma |} * f)^{p}}\, \,\mathrm {d}v\bigg ). \end{aligned}$$

(A.13)

Such an inequality is also easily deduced from (A.12) when $p=2$. We see then, thanks to Young’s inequality, that for all $r_3\geqslant 0$, there exists $r_4\geqslant 0$ such that

$$\begin{aligned}&\int _{{\mathbb {R}}^3} \langle v\rangle ^{r_3} f\,(1+f) \, ( |\cdot |^{- |\gamma |} * f)^p\, \,\mathrm {d}v \nonumber \\&\quad \leqslant \int _{{\mathbb {R}}^3} \langle v\rangle ^{r_4} f^2\,(1+f)^2 \, \,\mathrm {d}v\nonumber \\&\qquad + {2^{p-1}}\int _{{\mathbb {R}}^3} \langle v\rangle ^{-4} \, ( |\cdot |^{- |\gamma |}\, {\mathbf {1}}_{|\cdot |\leqslant 1} * f)^{2p}\, \,\mathrm {d}v\nonumber \\&\quad + {2^{p-1}} \int _{{\mathbb {R}}^3} \langle v\rangle ^{-4} \, ( |\cdot |^{- |\gamma |}\, {\mathbf {1}}_{|\cdot |\geqslant 1} * f)^{2p}\, \,\mathrm {d}v\nonumber \\&\quad \leqslant \int _{{\mathbb {R}}^3} \langle v\rangle ^{r_4} f^2\,(1+f)^2 \, \,\mathrm {d}v + C_p\, \Vert f\Vert _{L^{2p}}^{2p} + C\, \Vert f\Vert _{L^{1}}^{2p}\,, \end{aligned}$$

(A.14)

where we used that

$$\begin{aligned}&\int _{{\mathbb {R}}^3} \langle v\rangle ^{-4}\, (|\cdot |^{- |\gamma |}\, {\mathbf {1}}_{|\cdot |\leqslant 1} *f)^{2p}(v)\, \,\mathrm {d}v \leqslant \Vert \, |\cdot |^{- |\gamma |}\, {\mathbf {1}}_{|\cdot |\leqslant 1} *f \Vert _{L^{2p}}^{2p}\\&\leqslant \Vert \, |\cdot |^{- |\gamma |}\, {\mathbf {1}}_{|\cdot |\leqslant 1}\Vert _{L^{1}}^{2p} \, \Vert f\Vert _{L^{2p}}^{2p}\,, \end{aligned}$$

and $\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{-4}\,\mathrm {d}v < \infty$. Then,

$$\begin{aligned}&\int _{{\mathbb {R}}^3} \langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} (1+f)^2 \, |\partial _i f|^{p} \,\mathrm {d}v\\&\quad = - \int _{{\mathbb {R}}^3} \partial _i [ \langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} (1+f)^2 \, |\partial _i f|^{p-2} \, \partial _i f ]\, f \,\mathrm {d}v\\&\quad = - \int _{{\mathbb {R}}^3} \partial _i [\langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)}]\, (1+f)^2 \,f |\partial _i f|^{p-2} \, \partial _i f \, \,\mathrm {d}v \\&\qquad - 2\int _{{\mathbb {R}}^3} \langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} \,{f(1+f)}\, |\partial _i f|^{p} \,\mathrm {d}v \\&\qquad - (p-1) \int _{{\mathbb {R}}^3}\langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} (1+f)^2 \,f\, |\partial _i f|^{p-2} \, \partial _{ii} f \,\mathrm {d}v . \end{aligned}$$

The second integral is nonnegative whereas we can estimate the third integral using Young’s inequality to get, for any $\delta >0$,

$$\begin{aligned}&\int _{{\mathbb {R}}^3} \langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} {(1+f)^2} \, |\partial _i f|^{p} \,\mathrm {d}v\\&\quad \leqslant C \int _{{\mathbb {R}}^3} \langle v\rangle ^{m -1 + \max (|\gamma |, 2 - |\gamma |)} {(1+f)^2} \,f\, |\partial _i f|^{p-1} \,\mathrm {d}v\\&\qquad + \delta \int _{{\mathbb {R}}^3} \langle v\rangle ^{m -|\gamma |} \, |\partial _i f|^{p-2}\, |\partial _{ii} f|^2 \,\mathrm {d}v \\&\qquad + {\frac{(p-1)^2}{4\delta }} \int _{{\mathbb {R}}^{3}} {\langle v\rangle ^{m + \max (3|\gamma |, 4-|\gamma |)} (1+f)^4} \,f^2\, |\partial _i f|^{p-2} \,\mathrm {d}v. \end{aligned}$$

To estimate the first integral, we now use Young’s inequality in the form

$$\begin{aligned} \begin{aligned} \langle v\rangle ^{-1}{(1+f)^2} \,f\, |\partial _i f|^{p-1}&\leqslant \frac{1}{4}|\partial _{i}f|^{p}+ C_{p}\langle v\rangle ^{-p}(1+f)^{2p}f^{p} \\&\leqslant \frac{1}{4}(1+f)^{2}|\partial _{i}f|^{p}+ C_{p}\langle v\rangle ^{-p}(1+f)^{2p}f^{p}\,, \end{aligned} \end{aligned}$$

while, for the third integral, since $\max (3|\gamma |,4-|\gamma |) {\leqslant } \max (|\gamma |,2-|\gamma |)+2\max (|\gamma |,1)$, one can use Young’s inequality in the form

$$\begin{aligned}&\frac{(p-1)^{2}}{ {4}\delta }\langle v\rangle ^{2\max (|\gamma |,1)}(1+f)^{2}f^{2}|\partial _{i}f|^{p-2}\\&\leqslant \frac{1}{4}|\partial _{i}f|^{p} + C_{\delta ,p}\langle v\rangle ^{ {p} \, \max (|\gamma |,1)}(1+f)^{p}f^{p}\,, \end{aligned}$$

for some positive constant $C_{\delta ,p} >0$. Therefore, one can find $C_{\delta ,p}>0$ such that

$$\begin{aligned}&\int _{{\mathbb {R}}^3} \langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} {(1+f)^2} \, |\partial _i f|^{p} \,\mathrm {d}v \\&\quad \leqslant {\frac{1}{4}} \int _{{\mathbb {R}}^3} \langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} \, (1+f)^{2}|\partial _i f|^{p} \,\mathrm {d}v \\&\qquad + C \int _{{\mathbb {R}}^3} \langle v\rangle ^{r_5} {(1+f)^{2p}} \, f^p \,\mathrm {d}v + \delta \int _{{\mathbb {R}}^3} \langle v\rangle ^{m -|\gamma |} \, |\partial _i f|^{p-2}\, |\partial _{ii} f|^2 \,\mathrm {d}v\\&\qquad + {\frac{1}{4}} \int _{{\mathbb {R}}^3} \langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} {(1+f)^2} \, |\partial _i f|^{p} \,\mathrm {d}v\\&+ {C_{\delta ,p}} \int _{{\mathbb {R}}^3} \langle v\rangle ^{r_6} {(1+f)^{2 + p}} \, f^p \,\mathrm {d}v , \end{aligned}$$

with $r_{5}:=m + \max (|\gamma |, 2 - |\gamma |)-p$, $r_{6}:=m + \max (|\gamma |, 2 - |\gamma |) + p\,\max (|\gamma |,1).$ As a consequence,

$$\begin{aligned}&{\frac{1}{2}}\int _{{\mathbb {R}}^3} \langle v\rangle ^{m + \max (|\gamma |, 2 - |\gamma |)} (1+f)^2 \, |\partial _i f|^{p} \,\mathrm {d}v \nonumber \\&\quad \leqslant \delta \int _{{\mathbb {R}}^3} \langle v\rangle ^{m -|\gamma |} \, |\partial _i f|^{p-2}\, |\partial _{ii} f|^2 \,\mathrm {d}v\nonumber \\&\qquad + C_{\delta ,p} \int _{{\mathbb {R}}^3} \langle v\rangle ^{\max (r_5,r_6)} {(1+f)^{2 + 2p}} \, f^p \,\mathrm {d}v, \qquad \forall \delta >0. \end{aligned}$$

(A.15)

Using estimates (A.14), (A.15), (A.13), and remembering (A.6), we conclude the proof. $\square$

Notice that in particular, thanks to Proposition A.3, a Sobolev estimate in the v variable shows that f satisfies an $L^{\infty }$ (local w.r.t. time) estimate. More specifically, we have the following result

Corollary A.4

Assume $-2<\gamma <0$, and let $f_{\mathrm {in}}\in L^{\infty -0}({\mathbb {R}}^3) \cap L^1_{\infty }({\mathbb {R}}^{3})$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and let f(t, v) be a weak solution to (1.11.10). Then for any $p > 2$ and any $T >0$, if $f_{\mathrm {in}}\in W^{1,p}_{2}({\mathbb {R}}^3)$, there is some $C_T(p, f_{\mathrm {in}})$ depending on p, T and $f_{\mathrm {in}}$, such that

$$\begin{aligned} \left\| \, \varvec{\Sigma }[f]\, \nabla f - \varvec{b}[f]\, [f\,(1 - \varvec{\varepsilon }\,f)]\, \right\| _{L^{\infty }([0,T] ; L^p({\mathbb {R}}^3))} \leqslant C_T(p, f_{\mathrm {in}}) . \end{aligned}$$

(A.16)

Proof

Observe that thanks to (A.10) and (A.11),

$$\begin{aligned} \left| \langle v\rangle ^{-2-\gamma }\varvec{\Sigma }[f]\right| \leqslant C\,\int _{{\mathbb {R}}^{3}}\langle v_{*}\rangle ^{2+\gamma }f(v_{*})\, \mathrm {d}v_{*}{\leqslant C \Vert f_{\mathrm {in}}\Vert _{L^1_2}}\,, \end{aligned}$$

so that $\langle \cdot \rangle ^{-2-\gamma }\varvec{\Sigma }[f]$ is bounded in $L^{\infty }([0,T]\times {\mathbb {R}}^{3})$. In the same way $\langle \cdot \rangle ^{\min \{-1-\gamma ,0\}}\varvec{b}[f]$ is bounded in $L^{\infty }([0,T] \times {\mathbb {R}}^3)$. We conclude using Proposition A.3 and Corollary A.2. $\square$

Corollary A.5

Assume $-2<\gamma <0$ and let $f_{\mathrm {in}}\in L^{\infty -0}({\mathbb {R}}^3) \cap L^1_{\infty }({\mathbb {R}}^{3})$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and let f(t, v) be a weak solution to (1.11.10). Then for any $p > 2$, $T>0$, if $f_{\mathrm {in}}\in W^{1,p}_{2}({\mathbb {R}}^3)$ then there is some $C_T(p, f_{\mathrm {in}})$ depending on T, p and $f_{\mathrm {in}}$, such that

$$\begin{aligned} \Vert f \Vert _{W^{\frac{1}{3},p}([0,T] \times {\mathbb {R}}^3)} \leqslant C_T(p, f_{\mathrm {in}}) , \end{aligned}$$

(A.17)

and for any $\alpha \in (0, \frac{1}{3}), T >0$, there is some $C_T(\alpha , f_{\mathrm {in}})$ depending on T, $\alpha$ and $f_{\mathrm {in}}$, such that

$$\begin{aligned} \Vert f \Vert _{C^{0,\alpha }({[0,T]} \times {\mathbb {R}}^3)} \leqslant C_T(\alpha , f_{\mathrm {in}}). \end{aligned}$$

(A.18)

Proof

Using the equation and Corollary A.4, we see that, for all $1 \leqslant p < \infty$, if $f_{\mathrm {in}}\in W^{1,p}_{2}({\mathbb {R}}^3)$ then f is bounded in $W^{1,\infty }((0,T) ; W^{-1, p}({\mathbb {R}}^3))$. Proposition A.3 also ensures that f is bounded in $L^{\infty }((0,T); W^{1, p}({\mathbb {R}}^3))$. We get inequality (A.17) thanks to an interpolation, and deduce (A.18) from (A.17) thanks to a Sobolev inequality. $\square$

We now establish a weighted $L^{\infty }$ estimate which comes out of Prop. A.3.

Corollary A.6

Assume $-2<\gamma <0$ and let $f_{\mathrm {in}}\in L^{1}_{\infty }({\mathbb {R}}^3) {\cap L^{\infty -0}({\mathbb {R}}^3)}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and f(t, v) be a weak solution to (1.11.10). For any choice of $m \geqslant 0$ and $p > 3$, if $f_{\mathrm {in}}\in W^{1,p}_m({\mathbb {R}}^3)$, then there is some $C_T(m,p,f_{\mathrm {in}})$ depending on m, p, T and $f_{\mathrm {in}}$ such that

$$\begin{aligned} \sup _{t\in [0,T]} || \langle \cdot \rangle ^{\frac{m}{p}} \, f(t, \cdot ) ||_{L^{\infty }({\mathbb {R}}^3)} \leqslant C_T(m,p,f_{\mathrm {in}}) . \end{aligned}$$

(A.19)

Proof

Notice that, if $f_{\mathrm{in}} \in W^{1,p}({\mathbb {R}}^{3})$ with $p >3$, then $f_{\mathrm{in}} \in L^{\infty }$ thanks to Sobolev inequality. In particular, the assumptions of Propositions A.3 and Corollary A.2 are met. Using Sobolev inequality again, since $p >3$, there are $C_{0}=C_{0}(m,p),$ $C_{1}=C_{1}(m,p)$ such that

$$\begin{aligned}&\sup _{t\in [0,T]} \left\| \langle \cdot \rangle ^{\frac{m}{p}} \, f(t, \cdot ) \right\| _{L^{\infty }({\mathbb {R}}^3)} \leqslant C_{0} \sup _{t\in [0,T]} \left\| \langle \cdot \rangle ^{\frac{m}{p}} \, f(t, \cdot ) \right\| _{W^{1,p}({\mathbb {R}}^{3})}\\&\quad \leqslant C_{1} \sup _{t\in [0,T]}\left( \int _{{\mathbb {R}}^3} \langle v\rangle ^{m} \, | \nabla f(t,v) |^p \,\mathrm {d}v \right) ^{\frac{1}{p}}+ C_{1} \sup _{t\in [0,T]} \left( \int _{{\mathbb {R}}^3} \langle v\rangle ^{m} \, |f(t,v)|^p \,\mathrm {d}v\right) ^{\frac{1}{p}}. \end{aligned}$$

We conclude by using Proposition A.3 and Corollary A.2. $\square$

We can now state a stability estimate for the solutions of the LFD equation satisfying the regularity obtained in Proposition A.3.

Proposition A.7

Assume $-2<\gamma <0$ and let $f_{\mathrm {in}}, g_{\mathrm {in}}\in L^{1}_{\infty }({\mathbb {R}}^3) {\cap L^{\infty -0}({\mathbb {R}}^3)}$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and f(t, v), g(t, v) be weak solutions to (1.11.10). If $f_{\mathrm {in}}, g_{\mathrm {in}}\in W^{1,4}_{\infty }({\mathbb {R}}^3),$ then for all $m \geqslant 8$, there is some $C_T(m, f_{\mathrm {in}} , g_{\mathrm {in}})$ depending on m, T and $f_{\mathrm {in}}, g_{\mathrm {in}}$ such that

$$\begin{aligned}&\sup _{t\in [0,T]} \left\| \langle \cdot \rangle ^{\frac{m}{2}} \, (f(t, \cdot ) - g(t, \cdot )) \right\| _{L^{2}({\mathbb {R}}^3)}\nonumber \\&\leqslant C_T(m,f_{\mathrm {in}}, g_{\mathrm {in}})\, \left\| \langle \cdot \rangle ^{\frac{m}{2}} \, (f_{\mathrm {in}} - g_{\mathrm {in}}) \right\| _{L^{2}({\mathbb {R}}^3)} . \end{aligned}$$

(A.20)

Proof

In the rest of the proof, C will denote a positive constant depending on $m,T, {f_{\mathrm {in}}, g_{\mathrm {in}}}$ as well as the coercivity constant $K_{0}$ and that may change from one line to another. For two solutions f(t, v), g(t, v) to (1.11.10), we introduce

$$\begin{aligned} {\mathsf {u}}(t,v)=f(t,v)-g(t,v), \qquad {\mathsf {s}}(t,v)=f(t,v)+g(t,v) \end{aligned}$$

from which one sees easily that

$$\begin{aligned} \partial _{t} {\mathsf {u}}= & {} \frac{1}{2}\nabla \cdot \bigg (\left( \varvec{\Sigma }[f]+\varvec{\Sigma }[g]\right) \nabla {\mathsf {u}} -\varvec{b}[{\mathsf {u}}]\left( f(1-\varvec{\varepsilon }f)+g(1-\varvec{\varepsilon }g)\right) \\&+\varvec{\sigma }[{\mathsf {u}}(1-\varvec{\varepsilon }{\mathsf {s}})]\nabla {\mathsf {s}} - \varvec{b}[{\mathsf {s}}] {{\mathsf {u}}\left( 1-\varvec{\varepsilon }{\mathsf {s}}\right) }\bigg )\,. \end{aligned}$$

Using the short-hand notations ${\mathsf {h}}=f(1-\varvec{\varepsilon }f)+g(1-\varvec{\varepsilon }g),$ $\varvec{\Sigma }_{0}=\varvec{\Sigma }[f]+\varvec{\Sigma }[g]=\varvec{\sigma }[{\mathsf {h}}]$, we compute, for $m \geqslant 0$,

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m}{\mathsf {u}}^{2}(t,v)\,\mathrm {d}v= & {} -\int _{{\mathbb {R}}^{3}} \bigg (\varvec{\Sigma }_{0}\nabla {\mathsf {u}}\bigg ) \cdot \nabla \left( \langle v\rangle ^{m}{\mathsf {u}}(t,v)\right) \,\mathrm {d}v\\&+\int _{{\mathbb {R}}^{3}}{\mathsf {u}}(1-\varvec{\varepsilon }{\mathsf {s}}) \varvec{b}[{\mathsf {s}}] \cdot \nabla \left( \langle v\rangle ^{m}{\mathsf {u}}\right) \,\mathrm {d}v\\&+\int _{{\mathbb {R}}^{3}}{\mathsf {h}}\varvec{b}[{\mathsf {u}}] \cdot \nabla \left( \langle v\rangle ^{m}{\mathsf {u}}\right) \,\mathrm {d}v\\&-\int _{{\mathbb {R}}^{3}} \varvec{\sigma }[{\mathsf {u}}(1-\varvec{\varepsilon }{\mathsf {s}})]\nabla {\mathsf {s}} \cdot \nabla \left( \langle v\rangle ^{m}{\mathsf {u}}\right) \,\mathrm {d}v\\&=I_{1}+I_{2}+I_{3}+I_{4}\,. \end{aligned}$$

Here, arguing exactly as in [12,  Proof of Theorem 5.2, estimate of (5.2) + (5.4)], we can check that

$$\begin{aligned} I_{1}+I_{2}\leqslant & {} -K_{0}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }\left| \nabla {\mathsf {u}}\right| ^{2}\,\mathrm {d}v -\varvec{\varepsilon }\int _{{\mathbb {R}}^{3}}{\mathsf {u}} \langle v\rangle ^{m}{\mathsf {s}}\varvec{b}[{\mathsf {s}}]\cdot \nabla {\mathsf {u}}\,\mathrm {d}v \\&- \varvec{\varepsilon }m\int _{{\mathbb {R}}^{3}}{\mathsf {u}}^{2}\langle v\rangle ^{m-2}{\mathsf {s}} \varvec{b}[{\mathsf {s}}]\cdot v\,\mathrm {d}v\\&+\frac{m}{2}\int _{{\mathbb {R}}^{3}}{\mathsf {u}}^{2}\langle v\rangle ^{m-2}\varvec{b}[\mathsf {s+h}]\cdot v\,\mathrm {d}v +\frac{1}{2} \int _{{\mathbb {R}}^{3}}{\mathsf {u}}^{2}\varvec{\Sigma }_{0}:\nabla ^{2}\langle v\rangle ^{m}\,\mathrm {d}v\\&-\frac{1}{2}\int _{{\mathbb {R}}^{3}}{\mathsf {u}}^{2}\langle v\rangle ^{m}\varvec{c}[s]\,\mathrm {d}v\,. \end{aligned}$$

Observing that there is $C >0$ such that $\left| \varvec{\Sigma }_{0}\,:\nabla ^{2}\langle v\rangle ^{m}\right| \leqslant C\langle v\rangle ^{m+\gamma },$ while

$$\begin{aligned} \langle v\rangle ^{m-2}\varvec{b}[\mathsf {s+h}]\cdot v \leqslant C\left( \langle v\rangle ^{m+\gamma } - \langle v\rangle ^{m}\varvec{c}[\mathsf {s+h}]\right) \end{aligned}$$

thanks to (3.44), we can estimate the integrals involving $\int _{{\mathbb {R}}^{3}}{\mathsf {u}}^{2}\langle v\rangle ^{m} \varvec{c}[{\mathsf {s}}] \,\mathrm {d}v$ and $\int _{{\mathbb {R}}^{3}}{\mathsf {u}}^{2}\langle v\rangle ^{m}\varvec{c}[\mathsf {s+h}] \,\mathrm {d}v$ using Proposition 1.8 and deduce that

$$\begin{aligned} I_{1}+I_{2}\leqslant & {} -\frac{K_{0}}{2}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }\left| \nabla {\mathsf {u}}\right| ^{2}\,\mathrm {d}v +C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }{\mathsf {u}}^{2}\,\mathrm {d}v\\&- \varvec{\varepsilon }m \int _{{\mathbb {R}}^{3}}{\mathsf {u}}^{2}\langle v\rangle ^{m-2}{\mathsf {s}}\varvec{b}[{\mathsf {s}}]\cdot v\,\mathrm {d}v- \varvec{\varepsilon }\int _{{\mathbb {R}}^{3}} {\mathsf {u}}\langle v\rangle ^{m}{\mathsf {s}}\varvec{b}[{\mathsf {s}}]\cdot \nabla {\mathsf {u}}\,\mathrm {d}v\,. \end{aligned}$$

By Young’s inequality, the last term can be bounded by

$$\begin{aligned} \left| \varvec{\varepsilon }\int _{{\mathbb {R}}^{3}} {\mathsf {u}}\langle v\rangle ^{m}{\mathsf {s}}\varvec{b}[{\mathsf {s}}]\cdot \nabla {\mathsf {u}}\,\mathrm {d}v\right|\leqslant & {} \frac{K_{0}}{4}\int _{{\mathbb {R}}^{3}}\left| \nabla {\mathsf {u}}\right| ^{2}\langle v\rangle ^{m+\gamma }\,\mathrm {d}v\\&+C_{0}\,\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\,{\mathsf {s}}^{2}\left| \varvec{b }[{\mathsf {s}}]\right| ^{2}\,{\mathsf {u}}^{2}\,\mathrm {d}v\, \end{aligned}$$

for some positive constant $C_{0}$ depending on $K_{0}$. Using the bound in (A.19) together with the bound on $|\varvec{b}[s]|$ provided by (A.11), one checks without difficulty that there exists $C >0$ such that

$$\begin{aligned} {\mathsf {s}}| \varvec{b}[{\mathsf {s}}]| \leqslant C \langle v \rangle ^{ \gamma }, \end{aligned}$$

(A.21)

and therefore

$$\begin{aligned} I_{1}+I_{2}\leqslant & {} -\frac{K_{0}}{4}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }\left| \nabla {\mathsf {u}}\right| ^{2}\,\mathrm {d}v +C \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }{\mathsf {u}}^{2}\,\mathrm {d}v\\&+m \int _{{\mathbb {R}}^{3}}{\mathsf {u}}^{2}\langle v\rangle ^{m-2}\left| {\mathsf {s}}\varvec{b}[{\mathsf {s}}]\cdot v\right| \mathrm {d}v \end{aligned}$$

i.e.

$$\begin{aligned} I_{1}+I_{2} \leqslant -\frac{K_{0}}{2}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }\left| \nabla {\mathsf {u}}\right| ^{2}\,\mathrm {d}v +C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m}{\mathsf {u}}^{2}\,\mathrm {d}v \end{aligned}$$

(A.22)

where we used again (A.21). We now estimate $I_{3}$ and $I_{4}$. Since

$$\begin{aligned} I_{3}+I_{4}= & {} m\int _{{\mathbb {R}}^{3}}{\mathsf {u}}\langle v\rangle ^{m-2}{\mathsf {h}}\varvec{b}[{\mathsf {u}}]\cdot v\mathrm {d}v-m\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-2}{\mathsf {u}}\left( \varvec{\sigma }[\mathsf {u}(\mathsf {1}-\varvec{\varepsilon }\mathsf {s})]\nabla {\mathsf {s}}\right) \cdot v\,\mathrm {d}v\\&+\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m}{\mathsf {h}} \varvec{b}[{\mathsf {u}}]\cdot \nabla {\mathsf {u}}\mathrm {d}v-\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m}\left( \varvec{\sigma }[{\mathsf {u}}(1-\varvec{\varepsilon }{\mathsf {s}})]\nabla {\mathsf {s}}\right) \cdot \nabla {\mathsf {u}}\,\mathrm {d}v \end{aligned}$$

and, thanks to Young’s inequality,

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m}{\mathsf {h}} \varvec{b}[{\mathsf {u}}]\cdot \nabla {\mathsf {u}}\mathrm {d}v-\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m}\left( \varvec{\sigma }[{\mathsf {u}}(1-\varvec{\varepsilon }{\mathsf {s}})]\nabla {\mathsf {s}}\right) \cdot \nabla {\mathsf {u}}\,\mathrm {d}v\\&\quad \leqslant \frac{K_{0}}{4}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }\left| \nabla {\mathsf {u}}\right| ^{2}\,\mathrm {d}v + C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\left| \nabla {\mathsf {s}}\right| ^{2}\left| \,|\cdot |^{\gamma +2}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v\\&\qquad +C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\left| {\mathsf {h}}\right| ^{2}\left| \,|\cdot |^{\gamma +1}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v\,, \end{aligned}$$

while

$$\begin{aligned}&m\int _{{\mathbb {R}}^{3}}{\mathsf {u}}\langle v\rangle ^{m-2}{\mathsf {h}}\varvec{b}[{\mathsf {u}}] \cdot v\,\mathrm {d}v-m\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-2}{\mathsf {u}} \left( \varvec{\sigma }[\mathsf {u}(\mathsf {1}-\varvec{\varepsilon }\mathsf {s})]\nabla {\mathsf {s}}\right) \cdot v\,\mathrm {d}v\\&\quad \leqslant C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-1}{\mathsf {u}}^{2}\,\mathrm {d}v + C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-1}\left| {\mathsf {h}}\right| ^{2} \left| \,|\cdot |^{\gamma +1}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v\\&\qquad +C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-1}|\nabla {\mathsf {s}}|^{2} \,\left| \,|\cdot |^{\gamma +2}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v\,. \end{aligned}$$

Therefore,

$$\begin{aligned} I_{3}+I_{4}\leqslant & {} \frac{K_{0}}{4}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }\left| \nabla {\mathsf {u}}\right| ^{2}\,\mathrm {d}v +C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-1}{\mathsf {u}}^{2}\,\mathrm {d}v\\&+C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\left| \nabla {\mathsf {s}} \right| ^{2}\left| \,|\cdot |^{\gamma +2}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v\\&+C \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\left| {\mathsf {h}}\right| ^{2} \left| \,|\cdot |^{\gamma +1}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v\,. \end{aligned}$$

One has $|{\mathsf {h}}|^{2} \leqslant {\mathsf {s}}^{2}$ and, using again (A.19), there is $C >0$ such that $\langle v\rangle ^{m-\gamma +6}{\mathsf {h}}^{2}\leqslant C$, from which

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\left| {\mathsf {h}}\right| ^{2}\left| \,|\cdot |^{\gamma +1}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v \leqslant C \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{-6}\left| \,|\cdot |^{\gamma +1}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v. \end{aligned}$$

Splitting the convolution integral according to $|v-v_{*}| \geqslant 1$ and $|v-v_{*}| <1$, we see that

$$\begin{aligned} \left| \,|\cdot |^{\gamma +1}*{|{\mathsf {u}}|} \right| ^{2}\leqslant & {} 2\left( \int _{|v-v_{*}| \leqslant 1}|v-v_{*}|^{-1}|{\mathsf {u}}(v_{*})|\,\mathrm {d}v_{*}\right) ^{2}\\&+2\left( \int _{|v-v_{*}| >1}|v-v_{*}|\,|{\mathsf {u}}(v_{*})|\mathrm {d}v_{*}\right) ^{2} \end{aligned}$$

and, using Cauchy-Schwarz inequality, we see that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{-6}\left| \,|\cdot |^{\gamma +1}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v\leqslant & {} C\Vert {\mathsf {u}}\langle \cdot \rangle ^{3}\Vert _{L^{2}({\mathbb {R}}^{3})}^{2}\,\bigg (\int _{{\mathbb {R}}^{6}}\langle v\rangle ^{-6}\langle v_{*}\rangle ^{-6}|v-v_{*}|^{2}\,\mathrm {d}v\mathrm {d}v_{*}\\&+\int _{|v-v_{*}|\leqslant 1}\langle v\rangle ^{-6}\langle v_{*}\rangle ^{-6}|v-v_{*}|^{-2}\,\mathrm {d}v\,\mathrm {d}v_{*}\bigg ) \end{aligned}$$

from which

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\left| {\mathsf {h}}\right| ^{2}\left| \,|\cdot |^{\gamma +1}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v \leqslant {C}\left\| {\mathsf {u}}\,\langle \cdot \rangle ^{3}\right\| _{L^{2}({\mathbb {R}}^{3})}^{2}. \end{aligned}$$

(A.23)

Using now the estimate of $\nabla {\mathsf {s}}$ as deduced from Prop. A.3 in the form

$$\begin{aligned} \left\| \langle v\rangle ^{\frac{m - \gamma }{2} + 3} \nabla {\mathsf {s}}\right\| _{L^{4}({\mathbb {R}}^{3})} \leqslant C, \end{aligned}$$

we can apply twice Cauchy-Schwarz inequality to deduce that

$$\begin{aligned}&\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\left| \nabla {\mathsf {s}}\right| ^{2}\left| \,|\cdot |^{\gamma +2}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v\\&\quad \leqslant \bigg (\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{2m-2\gamma +12}\left| \nabla s\right| ^{4}\mathrm {d}v\bigg )^{\frac{1}{2}}\,\left( \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{-12}\left| \,|\cdot |^{\gamma +2}*{|{\mathsf {u}}|}\right| ^{4}\,\mathrm {d}v\right) ^{\frac{1}{2}}\\&\quad \leqslant C\bigg (\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{-12}\left( \int _{{\mathbb {R}}^{3}}|v-v_{*}|^{4+2\gamma }\langle v_{*}\rangle ^{-8}\,\mathrm {d}v_{*}\right) ^{2}\mathrm {d}v\bigg )^{\frac{1}{2}}\,\left\| {\mathsf {u}}\,\langle \cdot \rangle ^{4}\right\| _{L^{2}({\mathbb {R}}^{3})}^{2} \end{aligned}$$

from which

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-\gamma }\left| \nabla {\mathsf {s}}\right| ^{2}\left| \,|\cdot |^{\gamma +2}*{|{\mathsf {u}}|}\right| ^{2}\,\mathrm {d}v \leqslant C\left\| {\mathsf {u}}\langle \cdot \rangle ^{4}\right\| _{L^{2}({\mathbb {R}}^{3})}^{2}. \end{aligned}$$

Combining this with (A.23) we see that

$$\begin{aligned} I_{3}+I_{4}\leqslant \frac{K_{0}}{4}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m+\gamma }\left| \nabla {\mathsf {u}}\right| ^{2}\,\mathrm {d}v +C\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m-1}{\mathsf {u}}^{2}\,\mathrm {d}v + C\left\| {\mathsf {u}}\langle \cdot \rangle ^{4}\right\| _{L^{2}({\mathbb {R}}^{3})}^{2}. \end{aligned}$$

Combining this with the estimate (A.22) of $I_{1}+I_{2}$ we deduce that

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\int _{{\mathbb {R}}^{3}}\langle v\rangle ^{m}{\mathsf {u}}^{2}\,\mathrm {d}v \leqslant C\int _{{\mathbb {R}}^{3}}\big (\langle v\rangle ^{m}+\langle v\rangle ^{8}\big ){\mathsf {u}}^{2}\,\mathrm {d}v. \end{aligned}$$

Taking $m \geqslant 8$ and using Gronwall’s lemma, we get the stability estimate (A.7A.16A.20). $\square$

We conclude this Appendix with the proof that (for suitable initial data) the solutions of the LFD equation with moderately soft potentials are in fact classical.

Corollary A.8

Let $\gamma \in (-2,0)$. Consider an initial datum $f_{\mathrm {in}}\in L^{\infty -0}({\mathbb {R}}^3) \cap L^1_{\infty }({\mathbb {R}}^{3})\cap W^{1,p}_2({\mathbb {R}}^3)$ for some $p>2$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0 >0$. For any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_{0}]$, any weak solution f to (1.11.10) given by Theorem 1.5 is actually a classical solution, that is f is continuously differentiable with respect to t and twice continuously differentiable with respect to v on $(0, \infty )\times {\mathbb {R}}^3$.

Proof

We observe that f is a weak solution to the linear equation (with unknown u)

$$\begin{aligned} \partial _{t}u=\nabla \cdot \left( \varvec{\Sigma }[f] \nabla u \right) \, -(1-2\varvec{\varepsilon }\,f)\varvec{b}[f] \cdot \nabla u - \varvec{c}_\gamma [f]\,(1-\varvec{\varepsilon }f) u\,. \end{aligned}$$

Let $R>0$ and $\Omega =\{v\in {\mathbb {R}}^3 ; |v|\leqslant R\}$. The coefficients $\varvec{\Sigma }[f]$, $(1-2\varvec{\varepsilon }\,f)\varvec{b}[f]$, $\varvec{c}_\gamma [f]\,(1-\varvec{\varepsilon }\,f)$ and also $\nabla \varvec{\Sigma }[f]$ are Hölder-continuous on $(0,T) \times \Omega$ for any $T >0$ thanks to Corollary A.5 and belong to $L^\infty ((0,T) \times \Omega )$. We then deduce from Proposition 2.3 and [30, Chapter III, Theorem 12.1] that $\partial _tf$ and $\partial ^2_{v_i v_j} f$ are also Hölder-continuous on $(0,\infty ) \times \Omega$. $\square$

We now have all the ingredients for the

Proof of Theorem 1.6

The first statement in the Theorem is a direct consequence of Corollary A.2, while the second one is obtained thanks to Proposition A.3 and Corollary A.5 whenever $p > 2$. For $p \in [1,2]$, one deduces that $f \in L^{\infty }([0,T]; W^{1,p}_s({\mathbb {R}}^3))$ by a simple interpolation. The uniqueness part of the result is deduced from the stability estimate (A.7A.16A.20) in Proposition A.7. $\square$

Appendix B. About the Cauchy Theory

We give the detailed proof of Theorem 1.5 about the existence of solutions to (1.11.10). We follow the approach of [12]. Let $(\Psi _\nu )_{\nu \in (0,1)}$ be a family of smooth bounded functions on ${\mathbb {R}}_+$ that coincide with $\Psi (r)=r^{\gamma +2}$ for $0<\nu<r<\nu ^{-1}$ and satisfy that

• (i) The functions $\Psi _\nu '$, $\Psi _\nu ''$, $\Psi _\nu ^{(3)}$ and $\Psi _\nu ^{(4)}$ are bounded.

• (ii) The following hold

  $$\begin{aligned} \Psi _\nu (r)\geqslant \frac{\nu ^\gamma r^{2}}{2}\; \qquad \forall \, 0< r <\nu \,, \quad \Psi _\nu (r) \geqslant \frac{\nu ^{-(2+\gamma )}}{2}>0, \forall \, r >\nu ^{-1}\,. \end{aligned}$$

  (B.1)

• (iii) For any $r\in {\mathbb {R}}_+$,

  $$\begin{aligned} \Psi _\nu (r)\leqslant 2 \,r^{2+\gamma } \qquad \text { and } \qquad |\Psi '_\nu (r)|\leqslant C r^{1+\gamma }, \end{aligned}$$

  (B.2)

  for some constant C independent of $\nu$.

We then set

$$\begin{aligned} {\left\{ \begin{array}{ll} a^\nu (z) = \left( a^\nu _{i,j}(z)\right) _{i,j} \qquad \text{ with } \qquad a^\nu _{i,j}(z) = \Psi _\nu (|z|) \, \left( \delta _{i,j} -\frac{z_i z_j}{|z|^2} \right) , \\ b^\nu _i(z) = \sum _k \partial _k a^\nu _{i,k}(z) = - \, \frac{2 \, z_i}{|z|^2} \, \Psi _\nu (|z|), \\ c^\nu (z) =\sum _{k,l} \partial ^2_{kl} a^\nu _{k,l}(z) = - \, \frac{2}{|z|^2} \, \Big [\,\Psi _\nu (|z|) + |z| \,\Psi _\nu '(|z|)\,\Big ], \end{array}\right. } \end{aligned}$$

and we consider the following regularized problem

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _t f = \nabla \cdot \Big (\varvec{\Sigma }^{\nu }[f] \nabla f - \varvec{b}^{\nu }[f] f(1-\varvec{\varepsilon }f) \Big ) + \nu \Delta f\\ \displaystyle f(0,.)= f_{\mathrm {in}}\,, \end{array} \right. \end{aligned}$$

(B.3)

where, as above, $\varvec{\Sigma }^{\nu }[f]= a^{\nu }*(f(1-\varvec{\varepsilon }f))$, $\varvec{b}^{\nu }[f]=b^{\nu }*f$.

We note here that the initial condition of the regularized problem is not assumed to satisfy (1.9). For such an initial condition, Lemma 2.1 still holds. We first investigate the well-posedness of (B.3) and prove the following result.

Proposition B.1

Consider $f_{\mathrm {in}} \in {\mathscr {C}}^\infty ({\mathbb {R}}^3)\cap H^1({\mathbb {R}}^3) \cap W^{3,\infty }({\mathbb {R}}^3)$ such that

$$\begin{aligned} 0<\alpha _1 e^{-\beta _1 |v|^2} \leqslant f_{\mathrm {in}}(v) \leqslant \frac{ \alpha _2 \, e^{ -\beta _2 \, |v|^2}}{1+ \varvec{\varepsilon }\alpha _2 \, e^{-\beta _2 \, |v|^2}} \qquad \text{ for } \text{ every } v\in {\mathbb {R}}^3, \end{aligned}$$

(B.4)

for some positive constants $\alpha _1$, $\alpha _2$, $\beta _1$ and $\beta _2$. Let $\nu >0$ and $T>0$. Then, there exists a solution $f^\nu$ to the regularized problem (B.3) such that, for every $s>0$,

$$\begin{aligned} f^\nu \in L^\infty ((0,T);L^1_{s}({\mathbb {R}}^3))\cap L^2((0,T);H^1_{s}({\mathbb {R}}^3))\,. \end{aligned}$$

The proof of this Proposition can be easily adapted from the proof of [12, Theorem 4.2]. One begins by freezing the non-local coefficients in (B.3). The smoothness and boundedness of $\Psi _\nu$ are used here in order to obtain some parabolic operator with smooth coefficients and deduce the existence of a unique classical solution from [30,  Chapter V, Theorem 8.1]. Finally, some fixed-point argument enables to conclude.

In order to pass to the limit $\nu \rightarrow 0$ in (B.3) and obtain a solution to (1.11.10), we need to prove uniform estimates on $f^\nu$ (with respect to $\nu$). First, as in [12,  Lemma 4.8], one has the lemma:

Lemma B.2

For any $\sigma , t\in [0,T]$, $\sigma \leqslant t$, for any $\nu \in (0,1)$, the function $f^\nu$ satisfies

$$\begin{aligned}&\int _{{\mathbb {R}}^3}f^\nu (t,v) \,\mathrm {d}v = \varrho \,, \end{aligned}$$

(B.5)

$$\begin{aligned}&\int _{{\mathbb {R}}^3}f^\nu (t,v) |v|^2 \, \,\mathrm {d}v = {\theta } +6\nu t \varrho \leqslant {\theta } +6 T \varrho \,, \end{aligned}$$

(B.6)

$$\begin{aligned}&{{\mathcal {S}}}_{\varvec{\varepsilon }}( f_{\mathrm {in}}) \leqslant {{\mathcal {S}}}_{\varvec{\varepsilon }}(f^\nu (\sigma )) \; \leqslant \;{{\mathcal {S}}}_{\varvec{\varepsilon }}(f^\nu (t)) \,. \end{aligned}$$

(B.7)

where $\varrho = \displaystyle \int _{{\mathbb {R}}^3} f_{\mathrm {in}}(v)\,\mathrm {d}v$ and $\theta =\displaystyle \int _{{\mathbb {R}}^3} f_{\mathrm {in}}(v) \,|v|^2\,\mathrm {d}v$.

Next, we consider the ellipticity of the diffusion matrix. As in [12,  Proposition 4.9 and Corollary 4.10], one has the following proposition:

Proposition B.3

Let $0\leqslant f_{\mathrm {in}}\in L^{1}_{2}({\mathbb {R}}^{3})$ be fixed and satisfying (1.7) for some $\varvec{\varepsilon }_0 >0$. Let $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and $R(f_{\mathrm {in}})$ and $\eta (f_{\mathrm {in}})$ be given by the first point of Lemma 2.1. Let ${\overline{\eta }}$ be the constant given by the second point of Lemma 2.1 for $\delta =\eta (f_{\mathrm {in}})$. Let

$$\begin{aligned} 0<\nu \leqslant \min \left\{ (3R(f_{\mathrm {in}}))^{-1} , \left( \frac{3{\overline{\eta }}}{4\pi }\right) ^{\frac{1}{3}},1\right\} . \end{aligned}$$

Then,

• (1) there exists a positive constant $K_{0} > 0$ depending on $\gamma$, $\Vert f_{\mathrm {in}}\Vert _{L^1_2}$, and $H(f_{\mathrm {in}})$, such that, for any $v,\, \xi \in {\mathbb {R}}^3$,

  $$\begin{aligned}&\sum _{i,j} \, \left( \varvec{\Sigma }_{i,j}[f](v) +\nu \, \delta _{i,j} \right) \, \xi _i \, \xi _j \\&\quad \geqslant K_{0} \langle v \rangle ^{\gamma } \,\min \left\{ (\nu ^{-1}|v|)^{-\gamma }, 2^{-\gamma }, 2(\nu |v|)^{-(2+\gamma )}\right\} |\xi |^2 \end{aligned}$$

  holds for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$;

• (2) there exists a positive constant $\kappa > 0$ depending on $\gamma$, $\Vert f_{\mathrm {in}}\Vert _{L^1_2}$, and $H(f_{\mathrm {in}})$, such that

  $$\begin{aligned} \forall \, v,\, \xi \in {\mathbb {R}}^3, \qquad \sum _{i,j} \, \left( \varvec{\Sigma }_{i,j}[f](v) +\nu \, \delta _{i,j} \right) \, \xi _i \, \xi _j \geqslant \kappa \; \frac{ |\xi |^2}{1+|v|^2}, \end{aligned}$$

  holds for any $\varvec{\varepsilon }\in (0,\varvec{\varepsilon }_0]$ and $f \in {\mathcal {Y}}_{\varvec{\varepsilon }}(f_{\mathrm {in}})$.

The proof of the first point of this Proposition can be easily adapted from that of [24,  Proposition 2.3]. Indeed, $\Psi _\nu$may be bounded from below thanks to (B.1):

$$\begin{aligned} \Psi _\nu (r)\geqslant \min \left\{ \frac{\nu ^\gamma r^2}{2}, r^{2+\gamma }, \frac{\nu ^{-(2+\gamma )}}{2}\right\} \qquad \text{ for } \text{ any } r\in {\mathbb {R}}_+. \end{aligned}$$

The second point follows easily from the proof of the first point by using that $\nu ^{-1} \geqslant 3R(f_{\mathrm {in}})$. This gives some uniform (with respect to $\nu$) ellipticity estimate.

Lemma B.4

Let $f_{\mathrm {in}}\in {\mathscr {C}}^\infty ({\mathbb {R}}^3)\cap H^1({\mathbb {R}}^3) \cap W^{3,\infty }({\mathbb {R}}^3)$ satisfying (B.4). Let $f^\nu$ be a solution to (B.3) given by Proposition B.1. Then, for any $T>0$ and $s>2$, there exists some constant $C_s$ depending only on s, T and $\Vert f_{\mathrm {in}}\Vert _{L^1_2}$ such that

$$\begin{aligned} \sup _{t\in [0,T]} \Vert f^\nu (t)\Vert _{L^1_{s}} \leqslant \Vert f_{\mathrm {in}}\Vert _{L^1_{s}} \,\exp \left( {C_s T}\right) . \end{aligned}$$

(B.8)

Proof

With notations similar to those in (3.1), one has

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} \int _{{\mathbb {R}}^3} f^\nu (t,v)\, \langle v\rangle ^s \,\mathrm {d}v= & {} {\mathscr {J}}^\nu _{s,1}(f^\nu ,f^\nu )-\varvec{\varepsilon }{\mathscr {J}}^\nu _{s,1}(f^\nu ,(f^\nu )^2)\\&+{\mathscr {J}}^\nu _{s,2}(f^\nu ,F^\nu ) + s\nu \int _{{\mathbb {R}}^3} f^\nu \langle v \rangle ^{s-4} (3+(s+1)|v|^2) \, \,\mathrm {d}v, \end{aligned}$$

where

$$\begin{aligned} {\mathscr {J}}^\nu _{s,1}(h,g)&=2s\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}h(v)g(v_{*})\,\frac{\Psi _\nu (|v-v_{*}|)}{|v-v_{*}|^2} \left( \langle v\rangle ^{s-2}-\langle v_{*}\rangle ^{s-2}\right) \\&\quad \left( |v_{*}|^{2}-(v \cdot v_{*})\right) \mathrm {d}v\mathrm {d}v_{*}\,,\\ {\mathscr {J}}^\nu _{s,2}(h,g)&=s(s-2)\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}\langle v\rangle ^{s-4}h(v)g(v_{*})\frac{\Psi _\nu (|v-v_{*}|)}{|v-v_{*}|^2}\\&\quad \left( |v|^{2}\,|v_{*}|^{2}-(v\cdot v_{*})^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}\,. \end{aligned}$$

As in Lemma 3.1 and Remark 3.2 one has

$$\begin{aligned} {\mathscr {J}}^\nu _{s,1}(f^\nu ,f^\nu )=2s\int _{{\mathbb {R}}^{3}\times {\mathbb {R}}^{3}}f^\nu f_{*}^\nu \,\frac{\Psi _\nu (|v-v_{*}|)}{|v-v_{*}|^2} \langle v\rangle ^{s-2} \left( \langle v_{*}\rangle ^{2}-\langle v\rangle ^{2}\right) \,\mathrm {d}v\mathrm {d}v_{*}\leqslant 0. \end{aligned}$$

One now splits ${\mathscr {J}}^\nu _{s,2}(f^\nu ,F^\nu )$ according to $|v-v_{*}|\geqslant 1$ and $|v-v_{*}|< 1$ ,

$$\begin{aligned} {\mathscr {J}}^\nu _{s,2}(f^\nu ,F^\nu ) = I_1+I_2, \end{aligned}$$

where

$$\begin{aligned} I_1= & {} s(s-2) \int _{|v-v_{*}|\geqslant 1} f^\nu f_{*}^\nu (1-{\varvec{\varepsilon }}f_{*}^\nu ) \,\\&\times \frac{\Psi _\nu (|v-v_{*}|)}{|v-v_{*}|^2}\, \langle v\rangle ^{s-4} (|v|^2|v_{*}|^2 -(v\cdot v_{*})^2) \,\,\mathrm {d}v\, \mathrm {d}v_{*}, \\ I_2= & {} s(s-2) \int _{|v-v_{*}|< 1} f^\nu f_{*}^\nu (1-{\varvec{\varepsilon }}f_{*}^\nu )\\&\, \frac{\Psi _\nu (|v-v_{*}|)}{|v-v_{*}|^2}\, \langle v\rangle ^{s-4} (|v|^2|v_{*}|^2 -(v\cdot v_{*})^2) \,\,\mathrm {d}v\, \mathrm {d}v_{*}\,. \end{aligned}$$

Since $|v|^2 |v_{*}|^2 -(v\cdot v_{*})^2 \leqslant \langle v \rangle ^2 \langle v_{*}\rangle ^2$, $\Psi _\nu$ satisfies (B.2) and $F^\nu \leqslant f^\nu$, one has

$$\begin{aligned} I_1\leqslant & {} 2s(s-2) \int _{|v-v_{*}|\geqslant 1} |v-v_{*}|^\gamma f^\nu f_{*}^\nu \langle v \rangle ^{s-2} \langle v_{*}\rangle ^2 \,\mathrm {d}v \, \mathrm {d}v_{*}\\\leqslant & {} 2s(s-2) \int _{|v-v_{*}|\geqslant 1}f^\nu f_{*}^\nu \langle v \rangle ^{s-2} \langle v_{*}\rangle ^2 \,\mathrm {d}v \, \mathrm {d}v_{*}\leqslant 2 s(s-2) \varvec{m}_{s-2}^{\nu }(t) \varvec{m}_2^{\nu }(t), \end{aligned}$$

where $\varvec{m}_{s}^{\nu }(t) = \int _{{\mathbb {R}}^3} f^{\nu }(t,v) \langle v \rangle ^s \,\mathrm {d}v$. For $I_2$, we use (B.2), $F^\nu \leqslant f^\nu$, and $|v|^2|v_{*}|^2 -(v\cdot v_{*})^2 \leqslant |v|\, |v_{*}|\, |v-v_{*}|^2\,,$ to get that

$$\begin{aligned} I_2\leqslant & {} 2s(s-2) \int _{|v-v_{*}|<1} |v-v_{*}|^{2+\gamma } f^\nu f_{*}^\nu \langle v \rangle ^{s-4} |v| \, |v_{*}| \, \,\mathrm {d}v \, \mathrm {d}v_{*}\\\leqslant & {} 2s(s-2) \int _{|v-v_{*}|<1} f^\nu f_{*}^\nu \langle v \rangle ^{s-4} |v| \, |v_{*}| \, \,\mathrm {d}v \, \mathrm {d}v_{*}\leqslant 2s(s-2) \varvec{m}_{s-3}^{\nu }(t) \varvec{m}_1^{\nu }(t). \end{aligned}$$

One also splits ${\mathscr {J}}^\nu _{s,1}(f^\nu ,(f^\nu )^2)$ according to $|v-v_{*}|\geqslant 1$ and $|v-v_{*}|< 1$,

$$\begin{aligned} \varvec{\varepsilon }{\mathscr {J}}^\nu _{s,1}(f^\nu ,(f^\nu )^2) = J_1+J_2, \end{aligned}$$

where

$$\begin{aligned} J_1= & {} 2s \varvec{\varepsilon }\int _{|v-v_{*}|\geqslant 1} f^\nu \left( f_{*}^\nu \right) ^2\\&\, \frac{\Psi _\nu (|v-v_{*}|)}{|v-v_{*}|^2}\, (\langle v\rangle ^{s-2}- \langle v_{*}\rangle ^{s-2}) (v_{*}\cdot (v_{*}-v))\,\,\mathrm {d}v \,\mathrm {d}v_{*},\\ J_2= & {} 2s \varvec{\varepsilon }\int _{|v-v_{*}|<1} f^\nu \left( f_{*}^\nu \right) ^2 \,\\&\frac{\Psi _\nu (|v-v_{*}|)}{|v-v_{*}|^2}\, (\langle v\rangle ^{s-2}- \langle v_{*}\rangle ^{s-2}) (v_{*}\cdot (v_{*}-v))\,\,\mathrm {d}v \,\mathrm {d}v_{*}\,. \end{aligned}$$

Since $\Psi _\nu$ satisfies (B.2) and $\varvec{\varepsilon }f^\nu \leqslant 1$, one has

$$\begin{aligned} J_1\leqslant & {} 4s \int _{|v-v_{*}|\geqslant 1} |v-v_{*}|^\gamma f^\nu f_{*}^\nu \, {(\langle v\rangle ^{s-2}+ \langle v_{*}\rangle ^{s-2}) (\langle v_{*}\rangle ^2 +\langle v_{*}\rangle \langle v \rangle )}\,\,\mathrm {d}v \,\mathrm {d}v_{*}\\\leqslant & {} 4s\, ( \varvec{m}^\nu _{s-2}(t)\varvec{m}^\nu _2(t)+2\varvec{m}^\nu _{s-1}(t)\varvec{m}^\nu _1(t)+ {\varvec{m}^\nu _0(t)}\varvec{m}^\nu _s(t))\,. \end{aligned}$$

For $J_2$, we use that $\Psi _\nu (r)\leqslant 2$ for $r<1$ by (B.2) and $\varvec{\varepsilon }f^\nu \leqslant 1$. We also have $|v_{*}\cdot (v_{*}-v)|\leqslant |v_{*}| \, |v-v_{*}| ,$ and

$$\begin{aligned} \left| \langle v \rangle ^{s-2} - \langle v_{*}\rangle ^{s-2} \right| \leqslant (s-2) \,|v-v_{*}| \,\sup _{t\in (0,1)} \langle tv+(1-t)v_{*}\rangle ^{s-3} \,. \end{aligned}$$

Hence,

$$\begin{aligned} \left| \langle v \rangle ^{s-2} - \langle v_{*}\rangle ^{s-2} \right| \leqslant \left\{ \begin{array}{ll} (s-2)\,|v-v_{*}|\, &{}\quad \text{ if } s\leqslant 3\,, \\ C(s-2)\,|v-v_{*}|\, (\langle v\rangle ^{s-3} +\langle v_{*}\rangle ^{s-3} ) &{}\quad \text{ if } s>3\,, \end{array}\right. \end{aligned}$$

for some C depending only on s. Consequently, if $s\leqslant 3$, we obtain

$$\begin{aligned} J_2\leqslant 4s(s-2)\varvec{m}^\nu _0(t)\varvec{m}^\nu _1(t), \end{aligned}$$

whereas, if $s>3$,

$$\begin{aligned} J_2\leqslant 4C s(s-2)\big ( \varvec{m}^\nu _{s-3}(t)\varvec{m}^\nu _1(t) + \varvec{m}^\nu _0(t)\varvec{m}^\nu _{s-2}(t) \big ). \end{aligned}$$

Finally,

$$\begin{aligned} \nu s \int _{{\mathbb {R}}^3} f^\nu \, \langle v\rangle ^{s-4} (3+(1+s)|v|^2) \,\,\mathrm {d}v \leqslant \nu s (s+3)\varvec{m}^\nu _{s-2}(t). \end{aligned}$$

Combining the above estimates and (B.5)-(B.6), we deduce the existence of some constant $C_s$ depending on s, $\gamma$, T and $\Vert f_{\mathrm {in}}\Vert _{L^1_2}$ such that $\frac{\mathrm {d}}{\mathrm {d}t }\varvec{m}^\nu _s(t) \leqslant C_s \varvec{m}^\nu _s(t),$ and (B.8) follows. $\square$

Lemma B.5

Let $f_{\mathrm {in}}\in {\mathscr {C}}^\infty ({\mathbb {R}}^3)\cap H^1({\mathbb {R}}^3) \cap W^{3,\infty }({\mathbb {R}}^3)$ satisfying (B.4). Let $f^\nu$ be a solution to (B.3) given by Proposition B.1. Then, for any $T>0$ and $s\geqslant 2$, there exists some constant $C>0$ depending only on s, $\varvec{\varepsilon }$, T and $\Vert f_{\mathrm {in}}\Vert _{L^1_2}$ such that, for any $t\in (0,T)$,

(B.9)

with

Proof

Let $s\geqslant 0$. We deduce from (B.3) that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t} \int _{{\mathbb {R}}^3} (f^\nu )^2 \langle v \rangle ^s \,\mathrm {d}v= & {} -\int _{{\mathbb {R}}^3} \left( \varvec{\Sigma }^\nu [f^\nu ] +\nu I_3 \right) \nabla f^\nu \cdot \nabla f^\nu \langle v \rangle ^s \,\mathrm {d}v\nonumber \\&- s \int _{{\mathbb {R}}^3} \left( \varvec{\Sigma }^\nu [f^\nu ] f^\nu \nabla f^\nu \right) \cdot v \langle v \rangle ^{s-2} \,\mathrm {d}v\nonumber \\&+ \int _{{\mathbb {R}}^3} \varvec{b}^\nu [f^\nu ] \cdot \nabla f^\nu f^\nu (1-\varvec{\varepsilon }f^\nu ) \langle v \rangle ^s \,\mathrm {d}v\nonumber \\&+ s \int _{{\mathbb {R}}^3} \varvec{b}^\nu [f^\nu ] \cdot v \, (f^\nu )^2(1-\varvec{\varepsilon }f^\nu ) \langle v \rangle ^{s-2} \,\mathrm {d}v \nonumber \\&- s\,\nu \int _{{\mathbb {R}}^3} f^\nu \nabla f^\nu \cdot v\, \langle v \rangle ^{s-2} \,\mathrm {d}v\,. \end{aligned}$$

(B.10)

It follows from the second point of Proposition B.3 that

$$\begin{aligned} \int _{{\mathbb {R}}^3} \left( \varvec{\Sigma }^\nu [f^\nu ] +\nu I_3 \right) \nabla f^\nu \nabla f^\nu \langle v \rangle ^s \,\mathrm {d}v \geqslant \kappa \int _{{\mathbb {R}}^3} |\nabla f^\nu |^2 \, \langle v \rangle ^{s-2}\, \,\mathrm {d}v\,. \end{aligned}$$

Proceeding as in the proof of (3.12), we obtain

(B.11)

where $\varvec{A}(v)=\langle v\rangle ^{2}\mathbf {Id}+(s-2)\,v\otimes v$, $v \in {\mathbb {R}}^{3}.$ For the last two integrals in (B.11), we clearly have

and

For the integral involving $\varvec{A}$ in (B.11), we have by (B.2), for every i, j, $|\varvec{A}_{i,j}|\leqslant s\langle v\rangle ^2$ and $| \varvec{\Sigma }^\nu _{i,j}[f^\nu ]| \leqslant 2\Psi ^\nu *f^\nu \leqslant 4 |\cdot |^{2+\gamma } *f^\nu \,.$ Hence,

For the first integral in (B.11), since $0\leqslant \frac{1}{3} (f^\nu )^2\leqslant (f^\nu )^2- \frac{2\varvec{\varepsilon }}{3} (f^\nu )^3 \leqslant (f^\nu )^2$, we have, by (B.2),

$$\begin{aligned}&\left| s \int _{{\mathbb {R}}^3}\langle v\rangle ^{s-2} \left( (f^\nu )^2 -\frac{2\varvec{\varepsilon }}{3} (f^\nu )^3\right) \varvec{b}^\nu [f^\nu ] \cdot v \,\mathrm {d}v \right| \\&\quad \leqslant s \int _{{\mathbb {R}}^3}\langle v\rangle ^{s-2} (f^\nu )^2 \, | \varvec{b}^\nu [f^\nu ]|\, |v| \,\,\mathrm {d}v \\&\quad \leqslant 2s \int _{{\mathbb {R}}^6} \frac{ \Psi _\nu (|v-v_{*}|)}{|v-v_{*}|} \,f_{*}^\nu \, \langle v\rangle ^{s-1} \, (f^\nu )^2 \, \,\mathrm {d}v \mathrm {d}v_{*}\\&\quad \leqslant 4s \int _{{\mathbb {R}}^6} |v-v_{*}|^{1+\gamma } \,f_{*}^\nu \, \langle v\rangle ^{s-1} \, (f^\nu )^2 \, \,\mathrm {d}v \mathrm {d}v_{*}\,. \end{aligned}$$

Now, as for (3.42)-(3.43), there exists some universal constant $C>0$ such that for every $v\in {\mathbb {R}}^3$ and every $t\in [0,T]$,

$$\begin{aligned} \left| \int _{{\mathbb {R}}^3} |v-v_{*}|^{1+\gamma } \,f_{*}^\nu (t,v_{*})\mathrm {d}v_{*}\right|&\leqslant C (\Vert f_{\mathrm {in}}\Vert _{L^1_2} +\Vert {f^{\nu }(t)}\Vert _{L^2})\langle v\rangle ^{\max \{0,1+\gamma \}} \nonumber \\&\leqslant C (\Vert f_{\mathrm {in}}\Vert _{L^1_2} +\varvec{\varepsilon }^{-\frac{1}{2}} \Vert f_{\mathrm {in}}\Vert ^{\frac{1}{2}}_{L^1})\langle v\rangle ^{\max \{0,1+\gamma \}}, \end{aligned}$$

(B.12)

where we used that $f^\nu \leqslant \varvec{\varepsilon }^{-1}$. Consequently, we get that

Similarly, since $\varvec{\varepsilon }(f^\nu )^2 \leqslant f^\nu$, one has

For the second integral in (B.11), since $0\leqslant \frac{1}{6} (f^\nu )^2 \leqslant \frac{1}{2} (f^\nu )^2-\frac{\varvec{\varepsilon }}{3} (f^\nu )^3 \leqslant \frac{1}{2} (f^\nu )^2$, we have by (B.2)

$$\begin{aligned}&\left| -\int _{{\mathbb {R}}^3} \langle v \rangle ^s \left( \frac{1}{2} (f^\nu )^2 -\frac{\varvec{\varepsilon }}{3} (f^\nu )^3\right) \varvec{c}^\nu [f^\nu ] \,\mathrm {d}v \right| \\&\quad \leqslant \frac{1}{2} \int _{{\mathbb {R}}^3} \langle v \rangle ^s (f^\nu )^2 \, |\varvec{c}^\nu [f^\nu ]| \,\mathrm {d}v \\&\quad \leqslant \int _{{\mathbb {R}}^6}\left( \frac{ \Psi _\nu (|v-v_{*}|)}{|v-v_{*}|^2} + \frac{ |\Psi _\nu '(|v-v_{*}|)| }{|v-v_{*}|}\right) \, f_{*}^\nu \, \langle v \rangle ^s \,(f^\nu )^2 \, \,\mathrm {d}v \mathrm {d}v_{*}\\&\quad \leqslant (2+C) \int _{{\mathbb {R}}^6} |v-v_{*}|^\gamma \, f_{*}^\nu \, \langle v \rangle ^s \,(f^\nu )^2 \, \,\mathrm {d}v \mathrm {d}v_{*}\,. \end{aligned}$$

Now, for a given $v\in {\mathbb {R}}^3$, one has, thanks to the Hölder inequality,

$$\begin{aligned} \int _{{\mathbb {R}}^3} |v-v_{*}|^\gamma \, f_{*}^\nu \, \mathrm {d}v_{*}&\leqslant \int _{|v-v_{*}|\geqslant 1} |v-v_{*}|^\gamma \, f_{*}^\nu \, \mathrm {d}v_{*}+\int _{|v-v_{*}|<1} |v-v_{*}|^\gamma \, f_{*}^\nu \, \mathrm {d}v_{*}\\&\leqslant \Vert f^\nu (t)\Vert _{L^{1}} +\Vert f^\nu (t)\Vert _{L^{p}}\,\left( \int _{|v-v_{*}|<1}|v-v_{*}|^{q\gamma }\mathrm {d}v_{*}\right) ^{\frac{1}{q}} \\&\leqslant {\bar{C}} \left( \Vert f_{\mathrm {in}}\Vert _{L^{1}} + \varvec{\varepsilon }^{-\frac{p-1}{p}} \Vert f_{\mathrm {in}}\Vert _{L^{1}}^{\frac{1}{p}} \right) \end{aligned}$$

for $p>1$ such that $-\gamma q<3$ where $\frac{1}{p}+\frac{1}{q}=1$. Hence,

For $\nu \in (0,1)$ and $\gamma \in (-2,0)$, Lemma B.2 implies that all the above $L^1$-moments are bounded by some constant depending only on T and $\Vert f_{\mathrm {in}}\Vert _{L^1_2}$. Thus, gathering the above estimates, (B.9) follows.$\square$

Remark B.6

Performing the same manipulations as above but using the first point of Proposition B.3, one obtains

where again C depends on s, $\varvec{\varepsilon }$, T and $\Vert f_{\mathrm {in}}\Vert _{L^1_2}$.

Proof of Theorem 1.5

Let us fix $T>0$. Consider $f_{\mathrm {in}}\in L^1_{s_0}({\mathbb {R}}^3)$ for some $s_0>2$ satisfying (1.7)–(1.9) for some $\varvec{\varepsilon }_0>0$. Then, there exists a sequence of functions $( f_{\mathrm {in},k})_{k\geqslant 1}$ in ${\mathscr {C}}^\infty ({\mathbb {R}}^3)\cap H^1({\mathbb {R}}^3) \cap W^{3,\infty }({\mathbb {R}}^3)$ such that $( f_{\mathrm {in},k})_{k\geqslant 1}$ converges towards f in $L^1_{s_0}({\mathbb {R}}^3)$ and

$$\begin{aligned} \alpha '_k e^{-\beta '_k |v|^2}\leqslant f_{\mathrm {in},k} \leqslant \frac{\alpha _k e^{-\beta _k |v|^2}}{1+\varvec{\varepsilon }\alpha _k e^{-\beta _k |v|^2}}, \end{aligned}$$

for some positive constants $\alpha _k$, $\alpha '_k$, $\beta _k$ and $\beta '_k$.

For every $k\in {\mathbb {N}}_*$, we set $\nu _k=\frac{1}{k}$ and $f_k=f^{\nu _k}$, where $f^{\nu _k}$ denotes a solution to (B.3) with initial datum $f_{\mathrm {in},k}$ given by Proposition B.1. Since $(f_{\mathrm {in},k})_{k\geqslant 1}$ is bounded in $L^1_{s_0}({\mathbb {R}}^3)$, we deduce from Lemma B.4 that $(f_k)_{k\geqslant 1}$ is bounded in $L^2((0,T);L^1_{s_0}({\mathbb {R}}^3))$. We now apply Lemma B.5 with $s=s_0> 2$. Since $(f_{\mathrm {in},k})_{k\geqslant 1}$ is bounded in $L^1_{s_0}({\mathbb {R}}^3)\cap L^\infty ({\mathbb {R}}^3)$, it is bounded in $L^2_{s_0}({\mathbb {R}}^3)$ and we deduce that there exists some constant $C_{T,\varvec{\varepsilon }}$ depending on T, $\varvec{\varepsilon }$ and $\sup _{k\geqslant 1}\Vert f_{\mathrm {in},k}\Vert _{L^1_{s_0}}$ such that, for any $k\in {\mathbb {N}}_*$,

$$\begin{aligned} \sup _{t\in [0,T]} \Vert f_k(t)\Vert ^2_{L^2_{s_0}} +\int _0^T \int _{{\mathbb {R}}^3}\langle v\rangle ^{s_0-2} |\nabla f_k(t,v)|^2 \,\mathrm {d}v\, \mathrm {d}t \leqslant C_{T,\varvec{\varepsilon }}. \end{aligned}$$

Consequently, $(f_k)_{k\geqslant 1}$ is bounded in $L^2((0,T);H^1({\mathbb {R}}^3))$. We then deduce from the weak formulation associated to (B.3) that $(\partial _tf_k)_{k\geqslant 1}$ is bounded in $L^1((0,T);(W^{2,\infty }({\mathbb {R}}^3))')$ and thus, for $m\geqslant 4$, in $L^1((0,T);(H^m({\mathbb {R}}^3))')$. Now, for $m\geqslant 4$, we have

$$\begin{aligned} H^1({\mathbb {R}}^3)\cap L^1_{s_0}({\mathbb {R}}^3) \subset L^1({\mathbb {R}}^3)\subset (H^m({\mathbb {R}}^3))', \end{aligned}$$

the embedding of $H^1({\mathbb {R}}^3)\cap L^1_{s_0}({\mathbb {R}}^3)$ in $L^1({\mathbb {R}}^3)$ being compact. We may thus conclude from [31, Corollary 4] that $(f_k)_{k\geqslant 1}$ is relatively compact in the space $L^2((0,T);L^1({\mathbb {R}}^3))$. Therefore, there exists a function $f\in L^2((0,T);L^1({\mathbb {R}}^3))$ and a subsequence of $(f_k)_{k\geqslant 1}$ (not relabelled) such that $(f_k)_{k\geqslant 1}$ converges towards $f\in L^2((0,T);L^1({\mathbb {R}}^3))$ and a.e. on $(0,T)\times {\mathbb {R}}^3$. For $\varphi \in {\mathscr {C}}_0^2({\mathbb {R}}^3)$, it is easy to check that the sequence $(\int _{{\mathbb {R}}^3} f_k \varphi \, \,\mathrm {d}v)_{k\geqslant 1}$ is equicontinuous and bounded in ${\mathscr {C}}([0,T])$. The Arzelà-Ascoli Theorem thus ensures that it is relatively compact in ${\mathscr {C}}([0,T])$.

Finally, we obtain that $(\int _{{\mathbb {R}}^3} f_k \varphi \, \,\mathrm {d}v)_{k\geqslant 1}$ converges towards $\int _{{\mathbb {R}}^3} f \varphi \, \,\mathrm {d}v$ in ${\mathscr {C}}([0,T])$ and then that $(f_k)_{k\geqslant 1}$ converges towards f in ${\mathscr {C}}_w([0,T];L^2({\mathbb {R}}^3))$, where ${\mathscr {C}}_w([0,T];L^2({\mathbb {R}}^3))$ denotes the space of weakly continuous functions in $L^2({\mathbb {R}}^3)$. We easily check that f preserves mass and energy and, passing to the limit $k\rightarrow \infty$ in the weak formulation, we obtain that f satisfies (1.11). Moreover, we can deduce from (B.8) that $f\in L^\infty ((0,T);L^1_{s_0}({\mathbb {R}}^3))$, and from Remark B.6 that $\nabla f \in L^{2}((0,T);L^{2}_{s_0+\gamma }({\mathbb {R}}^{3})).$

Let us now prove the monotonicity of the entropy. We know that $f\in L^2((0,T); H^1_{2+\gamma }({\mathbb {R}}^3))$. Then, we deduce as in (B.12) that there exists some constant $C_{\varvec{\varepsilon }}$ depending on $\varvec{\varepsilon }$ and $\Vert f_{\mathrm {in}}\Vert _{L^1_2}$ such that, for every $v\in {\mathbb {R}}^3$ and $t\in [0,T]$

$$\begin{aligned} \big |\varvec{b}[f](t,v) \big |\leqslant C_{\varvec{\varepsilon }}\, \langle v \rangle ^{\max \{0,1+\gamma \}}. \end{aligned}$$

One also has, for every $v\in {\mathbb {R}}^3$ and every $t\in [0,T]$, that $|\varvec{\Sigma }[f](t,v)| \leqslant C\Vert f_{\mathrm {in}}\Vert _{L^1_2} \,\langle v \rangle ^{2+\gamma },$ for some universal constant $C>0$. It thus follows from the weak formulation associated to (1.11.10) that $\partial _t f\in L^2((0,T);(H^1_{2+\gamma }({\mathbb {R}}^3))')$. We then deduce from [32, Ch.III Lemma 1.2] that $f\in {\mathscr {C}}([0,T];L^2({\mathbb {R}}^3))$. As in [12, Lemma 4.18], one may then prove the monotonicity of ${\mathcal {S}}_{\varvec{\varepsilon }}(f)$. $\square$