Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility

Abstract

We consider the sharp interface limit of a Navier–Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility $m_\varepsilon =\sqrt{\varepsilon }$, where the small parameter $\varepsilon >0$ related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier–Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen–Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable $\varepsilon$-scaled and coupled model problem. Moreover, we apply the novel idea of introducing $\varepsilon$-dependent coordinates.

1 Introduction and main result

Two-phase flows of macroscopically immiscible fluids is an important research area with many applications. There are two important model categories: sharp interface models and diffuse interface models. For sharp interface models the interface separating the fluids is assumed to be a hypersurface. These models usually consist of an evolution law for the hypersurface, coupled to equations in the bulk domains and on the interface. Solutions often develop singularities in finite time, in particular when the interface changes its topology. In contrast, diffuse interface models use a typically smooth order parameter (e.g. the density or volume fraction of the two fluids) that distinguishes the bulk domains and inbetween typically has steep gradients in a small transition zone (also called diffuse interface), which is proportional to a small parameter, e.g. $\varepsilon >0$. In applications the diffuse interface can be interpreted as microscopically small mixing region of the fluids. Quantities defined on the hypersurface in sharp interface models typically have a diffuse analogue that is defined in the diffuse interface. An important example is the relation of surface tension and capillary stress tensor, see Anderson et al. [9]. Diffuse interface models may be more suited to describe phenomena acting on length scales related to the interface thickness, e.g. interface thicking phenomena, complicated contact angle behaviour and topology changes, cf. [9]. Moreover, topology changes typically are no problem from an analytical or numerical point of view in contrast to sharp interface models. However, both model types are usually derived from physical principles or observations and can be used to model the same situations in applications. This motivates to study the connection between diffuse and sharp interface models by sending the small parameter $\varepsilon$ (related to the thickness of the diffuse interface) to zero. Such limits are known as “sharp interface limits”.

Let $T_0>0$, $\Omega \subseteq \mathbb {R}^2$ be a bounded smooth domain and $\varepsilon >0$ be small. For $\textbf{v}_\varepsilon :\overline{\Omega }\times [0,T_0]\rightarrow \mathbb {R}^2$, $p_\varepsilon , c_\varepsilon :\overline{\Omega }\times [0,T_0]\rightarrow \mathbb {R}$ we consider the following Navier–Stokes/Allen–Cahn system for small $\varepsilon >0$:

$$\begin{aligned} \partial _t \textbf{v}_\varepsilon +\textbf{v}_\varepsilon \cdot \nabla \textbf{v}_\varepsilon -{\text {div}}(2\nu (c_\varepsilon )D\textbf{v}_\varepsilon ) +\nabla p_\varepsilon&= -\varepsilon {\text {div}}(\nabla c_\varepsilon \otimes \nabla c_\varepsilon )&\quad&\text {in}\ \Omega \times (0,T_0),\end{aligned}$$

(1.1)

$$\begin{aligned} {\text {div}}\textbf{v}_\varepsilon&= 0&\quad&\text {in}\ \Omega \times (0,T_0),\end{aligned}$$

(1.2)

$$\begin{aligned} \partial _t c_\varepsilon +\textbf{v}_\varepsilon \cdot \nabla c_\varepsilon&=m_\varepsilon \left[ \Delta c_\varepsilon - \frac{1}{\varepsilon ^{2}} f'(c_\varepsilon )\right]&\quad&\text {in}\ \Omega \times (0,T_0),\end{aligned}$$

(1.3)

$$\begin{aligned} (\textbf{v}_\varepsilon ,c_\varepsilon )|_{\partial \Omega }&= (0,-1)&\quad&\text {on }\partial \Omega \times (0,T_0),\end{aligned}$$

(1.4)

$$\begin{aligned} (\textbf{v}_\varepsilon ,c_\varepsilon ) |_{t=0}&= (\textbf{v}_{0,\varepsilon },c_{0,\varepsilon })&\quad&\text {in }\Omega , \end{aligned}$$

(1.5)

where $\textbf{v}_\varepsilon , p_\varepsilon$ have the interpretation of a mean fluid velocity and pressure, respectively, and $c_\varepsilon$ has the role of an order parameter distinguishing two components of a fluid mixture. Moreover, $\nu :\mathbb {R}\rightarrow (0,\infty )$ is a smooth concentration-dependent viscosity, $m_\varepsilon :=\sqrt{\varepsilon }$ is the mobility and $f:\mathbb {R}\rightarrow [0,\infty )$ is a suitable smooth double-well potential with wells of equal depth, e.g. $f(c)=\frac{1}{8}(c^2-1)^2$, specified below. For simplicity in the following analysis we assume that $\nu ':\mathbb {R}\rightarrow \mathbb {R}$ is even. Furthermore, $D\textbf{v}_\varepsilon =\frac{1}{2}(\nabla \textbf{v}_\varepsilon +(\nabla \textbf{v}_\varepsilon )^T)$ is the symmetrized gradient and the operators $\nabla$, $\Delta$ and ${\text {div}}$ are defined to act on spatial variables only. Finally, note that $\nabla c_\varepsilon \otimes \nabla c_\varepsilon$ is a contribution to the stress tensor that represents capillary stresses due to surface tension effects in the (typically small) mixing region. The above model was introduced by Liu and Shen in [26] for constant viscosity along with a Navier–Stokes/Cahn-Hilliard variant in order to describe two-phase incompressible Newtonian fluids with the diffuse interface approach. The model was later derived in a thermodynamically consistent way by Jiang et al. [21] via an energetic variational approach including the case of different densities. Moreover, they showed global existence of weak solutions in 3D and global well-posedness and longtime behaviour of strong solutions in 2D.

We are interested in the sharp interface limit $\varepsilon \rightarrow 0$ for the above system (1.1)–(1.5). For well-prepared initial data and small times, we will rigorously prove the convergence of (1.1)–(1.5) to the following classical two-phase Navier–Stokes equation with surface tension:

$$\begin{aligned} \partial _t \textbf{v}_0^\pm +\textbf{v}_0^\pm \cdot \nabla \textbf{v}_0^\pm -\nu ^\pm \Delta \textbf{v}_0^\pm +\nabla p^\pm _0&= 0&\qquad&\text {in }\Omega ^\pm _t, t\in [0,T_0],\end{aligned}$$

(1.6)

$$\begin{aligned} {\text {div}}\textbf{v}_0^\pm&= 0&\qquad&\text {in }\Omega ^\pm _t, t\in [0,T_0],\end{aligned}$$

(1.7)

$$\begin{aligned} \llbracket 2\nu ^\pm D\textbf{v}_0^\pm -p_0^\pm \textbf{I}\rrbracket \textbf{n}_{\Gamma _t}&= -\sigma H_{\Gamma _t}\textbf{n}_{\Gamma _t}{} & {} \text {on }\Gamma _t, t\in [0,T_0],\end{aligned}$$

(1.8)

$$\begin{aligned} \llbracket \textbf{v}_0^\pm \rrbracket&=0{} & {} \text {on }\Gamma _t, t\in [0,T_0],\end{aligned}$$

(1.9)

$$\begin{aligned} V_{\Gamma _t}&=\textbf{n}_{\Gamma _t}\cdot \textbf{v}_0^\pm{} & {} \text {on }\Gamma _t, t\in [0,T_0],\end{aligned}$$

(1.10)

$$\begin{aligned} \textbf{v}_0^-|_{\partial \Omega }&= 0{} & {} \text {on }\partial \Omega \times (0,T_0),\end{aligned}$$

(1.11)

$$\begin{aligned} \Gamma _0=\Gamma ^0,\qquad \textbf{v}_0^\pm |_{t=0}&= \textbf{v}_{0,0}^\pm{} & {} \text { in }\Omega ^\pm _0, \end{aligned}$$

(1.12)

where $T_0>0$, $\nu ^\pm :=\nu (\pm 1)$, $\Omega$ is the disjoint union of $\Omega ^+_t, \Omega ^-_t$ and $\Gamma _t$ for every $t\in [0,T_0]$, $\Omega ^\pm _t$ are smooth domains, $\Gamma _t=\partial \Omega ^+_t\subseteq \Omega$ and $\textbf{n}_{\Gamma _t}$ is the interior normal of $\Gamma _t$ with respect to $\Omega ^+_t$. The jump $\llbracket u\rrbracket (.,t)$ in $x\in \Gamma _t$ of a quantity u defined on $\Omega ^+_t\cup \Omega ^-_t$ is defined as

$$\begin{aligned} \llbracket u\rrbracket (x,t)= \lim _{r\rightarrow 0+} \left[ u(x+r\textbf{n}_{\Gamma _t}(x),t)- u(x-r\textbf{n}_{\Gamma _t}(x),t)\right] . \end{aligned}$$

Moreover, $H_{\Gamma _t}$ is the (mean) curvature and $V_{\Gamma _t}$ is the normal velocity of $\Gamma _t$ with respect to $\textbf{n}_{\Gamma _t}$. Furthermore, $(\Gamma ^0,\textbf{v}_{0,0}^\pm )$ are suitable initial data. For the following we denote

$$\begin{aligned} \Omega ^\pm = \bigcup _{t\in [0,T_0]}\Omega ^\pm _t\times \{t\},\quad \Gamma = \bigcup _{t\in [0,T_0]}\Gamma _t\times \{t\}. \end{aligned}$$

The surface tension constant $\sigma$ is determined by $\sigma = \int _{\mathbb {R}}\theta _0'(\rho )^2\textrm{d}\rho$, where $\theta _0$ is the well-known optimal profile, i.e., the unique solution of

$$\begin{aligned} -\theta _0''+f'(\theta _0)=0\quad \text { in }\mathbb {R}, \qquad \lim _{\rho \rightarrow \pm \infty }\theta _0(\rho )=\pm 1, \qquad \theta _0(0)=0. \end{aligned}$$

(1.13)

As in Abels and Liu [5] we assume for the double well potential $f:\mathbb {R}\rightarrow \mathbb {R}$ that it is smooth and satisfies the assumptions

$$\begin{aligned} f'(\pm 1)=0, \quad f''(\pm 1)>0,\quad f(s)=f(-s)>0 \quad \text {for all }s\in (-1,1). \end{aligned}$$

Then there is a unique solution $\theta _0 :\mathbb {R}\rightarrow \mathbb {R}$ of (1.13), which is monotone. Moreover, for every $m\in \mathbb {N}_0$, there is some $C_m>0$ such that

$$\begin{aligned} |\partial _\rho ^m(\theta _0(\rho )\mp 1)|\le C_m e^{-\alpha |\rho |}\quad \text {for all } \rho \in \mathbb {R}\text { with }\rho \gtrless 0, \end{aligned}$$

where $\alpha =\min (\sqrt{f''(-1)}, \sqrt{f''(1)})$. Since f is assumed to be even, $\theta _0$ is odd and $\theta '_0$ is even.

Strong solutions for the problem (1.6)–(1.12) have been studied extensively in the literature starting with the results by Denisova and Solonnikov [14]. For further references we refer to Köhne et al. [24] and the monograph by Prüss and Simonett [30], where in particular local well-posedness in an $L^p$-setting is shown. Existence of a notion of weak solutions, called varifold-solutions, globally in time was shown in [1]. Weak-strong uniqueness for these kind of solutions was shown by Hensel and Fischer [16].

Let us now comment on the choice of vanishing mobility $m_\varepsilon =\sqrt{\varepsilon }\rightarrow 0$ in (1.4). In [2] a non-convergence result was shown for a convective Allen–Cahn equation for a mobility $m_\varepsilon =m_0\varepsilon ^\alpha$, where $m_0>0$ is a constant and $\alpha >2$, and formal asymptotic calculations were carried out for the case $\alpha =0,1$. Hence for constant mobility $m_\varepsilon =m_0$ the formal limit is a transport equation coupled to mean curvature flow, whereas for the case $m_\varepsilon =m_0\varepsilon$ the formal limit is a pure transport equation, cf. (1.10) above. It is possible to adapt the formal calculations to the case of mobilities $m_\varepsilon =m_0\varepsilon ^\alpha$ for all exponents $\alpha \in [0,1]$ (with the same limit system for $\alpha \in (0,1]$), the expansions just become more tedious and lengthy due to the fractional order ansatz. In Abels and Fei [3] the case of $m_\varepsilon =1$, $\alpha =0$ was studied and rigorous convergence to a two-phase Navier–Stokes system coupled with mean curvature flow was shown as expected from the formal asymptotic expansions as long as a smooth solution of the limit system exists. However, for this limit system there is no conservation of mass and hence it could be considered physically less relevant compared to the classical two-phase Navier–Stokes system with surface tension, where one has pure transport of the interface. This clearly motivates the study of the case of vanishing mobility $m_\varepsilon$ for $\varepsilon \rightarrow 0$. To the best of our knowledge there is no rigorous convergence result in the case of vanishing mobility in the literature. The choice of $m_\varepsilon =\sqrt{\varepsilon }$ and $\alpha =\frac{1}{2}$ for our result is motivated as follows: for the arguments in [3], the exponent $\alpha =\frac{1}{2}$ is critical in a heuristic sense by calculating the orders for $\alpha =1$ and by assuming some linear depencence on $\alpha$. The cases $\alpha \in (0,\frac{1}{2}]$ should work formally with the strategy in [3], but we decided to simply choose $\alpha =\frac{1}{2}$, in particular in order to have simpler asymptotic expansions with just $\sqrt{\varepsilon }$-spacing in the sums. We note that in a joint-work with Fischer the first and third author show convergence for more general scalings of $m_\varepsilon >0$ using the relative entropy method. In this work the convergence is obtained in weaker norms (and assuming same viscosities for simplicity), but it also holds for three space dimensions, see [4].

Our strategy to prove the sharp interface limit is via rigorous asymptotic expansions. The method goes back to de Mottoni and Schatzman [13] who first applied it to prove the rigorous sharp interface limit for the Allen–Cahn equation. The strategy works as follows: it is assumed that there exists a smooth solution to the limit sharp interface problem locally in time (usually this is no restriction). Then in the first step, one rigorously constructs an approximate solution to the diffuse interface system via rigorous asymptotic expansions based on the evolving hypersurface that is part of the solution to the limit problem. In the second step, one estimates the difference between the exact and approximate solution with the aid of a spectral estimate for a linear operator depending on the diffuse interface equation and the approximate solution. Comparison principles are not needed for the method and one even obtains the typical profile of solutions across the diffuse interface. The strategy was applied to many other sharp interface limits as well, see Moser [28] for a list of results. Let us just mention the famous result by Alikakos et al. [8] for the Cahn-Hilliard equation, Abels and Liu [5] for a Stokes/Allen–Cahn system, Abels and Marquardt [6, 7] for a Stokes/Cahn–Hilliard system, and the recent result Abels and Fei [3] for the Navier–Stokes/Allen Cahn system with constant mobility.

In general, rigorous results for sharp interface limits can be grouped into results concerning strong solutions for the limit system, in particular before singularities appear, and global time results using some weak notion for the sharp interface system. As described above, our result relies on the existence of a smooth solution for the limit system and assumes sufficiently small times. Another important strategy for sharp interface limits using strong solutions is the relative entropy method, see Fischer et al. [17] where the convergence of the Allen–Cahn-equation to mean curvature flow is considered and Hensel and Liu [19], where the Navier–Stokes/Allen–Cahn system with constant mobility (but equal viscosities) is considered, cf. [28] for more references concerning the relative entropy method. Weak notions used for global time results for the Allen–Cahn equation are viscosity solutions [10, 11, 15, 23], varifold solutions [20, 22, 27], BV-solutions ([25]; conditional result) and a solution concept inbetween [18]. In [3] there are more references for results on Navier–Stokes/Cahn–Hilliard-type models.

The following theorem is our main result about convergence of (1.1)–(1.5) to (1.6)–(1.12):

Theorem 1.1

Let $T_0>0$, $m_\varepsilon =\sqrt{\varepsilon }$ for all $\varepsilon >0$, and $(\textbf{v}_0^\pm ,\Gamma )$ be a smooth solution of the two-phase Navier–Stokes system with surface tension (1.6)–(1.12) on $[0,T_0]$ with $c_{0,\varepsilon }(x,t)\in [-1,1]$ for all $(x,t)\in \overline{\Omega }\times [0,T_0]$, $\varepsilon \in (0,1]$. Let $N\in \mathbb {N}$, $N\ge 3$. Then there exist $c_A=c_A(N,\varepsilon ),\textbf{v}_A=\textbf{v}_A(N,\varepsilon )\in H^1(0,T_0;L^2(\Omega ))\cap L^2(0,T_0;H^2(\Omega ))$ for $\varepsilon \in (0,1]$, uniformly bounded in these spaces and $\Vert c_A\Vert _\infty \le 1+c$ with $c>0$ independent of $\varepsilon \in (0,1]$, such that the following holds:

Let $(\textbf{v}_\varepsilon ,c_\varepsilon )$ be strong solutions of (1.1)–(1.5) with initial values $\textbf{v}_{0,\varepsilon }$, $c_{0,\varepsilon }$ such that

$$\begin{aligned} \Vert c_{0,\varepsilon }-c_A|_{t=0}\Vert _{L^2(\Omega )}+ \varepsilon ^2\Vert \nabla (c_{0,\varepsilon }-c_A|_{t=0})\Vert _{L^2(\Omega )}+ \Vert \textbf{v}_{0,\varepsilon } -\textbf{v}_A|_{t=0}\Vert _{L^2(\Omega )}\le C\varepsilon ^{N+\frac{1}{2}}\nonumber \\ \end{aligned}$$

(1.14)

for all $\varepsilon \in (0,1]$ and some $C>0$. Then there are some $\varepsilon _0 \in (0,1]$, $R>0$ and $T_1\in (0,T_0]$ small such that for all $\varepsilon \in (0,\varepsilon _0]$ and some $C_R>0$ it holds

$$\begin{aligned} \Vert c_\varepsilon -c_A\Vert _{L^\infty (0,T_1;L^2(\Omega ))}+\varepsilon ^{\frac{1}{4}}\Vert \nabla (c_\varepsilon -c_A)\Vert _{L^2((\Omega \times (0,T_1))\setminus \Gamma (\delta ))}&\le R\varepsilon ^{N+\frac{1}{2}},\end{aligned}$$

(1.15)

$$\begin{aligned} \varepsilon ^{\frac{1}{4}} \Vert \nabla _{{\varvec{\tau }}_\varepsilon }(c_\varepsilon -c_A)\Vert _{L^2((\Omega \times (0,T_1))\cap \Gamma (2\delta ))}+ \varepsilon \Vert \nabla (c_\varepsilon -c_A)\Vert _{L^2((\Omega \times (0,T_1))\cap \Gamma (2\delta ))}&\le R\varepsilon ^{N+\frac{1}{2}},\end{aligned}$$

(1.16)

$$\begin{aligned} \varepsilon ^2\Vert \nabla (c_\varepsilon -c_A)\Vert _{L^\infty (0,T_1;L^2(\Omega ))}+\varepsilon ^{2+\frac{1}{4}}\Vert \nabla ^2(c_\varepsilon -c_A)\Vert _{L^2(\Omega \times (0,T_1))}&\le R\varepsilon ^{N+\frac{1}{2}},\end{aligned}$$

(1.17)

$$\begin{aligned} \Vert \textbf{v}_\varepsilon -\textbf{v}_A\Vert _{H^{\frac{1}{2}}(0,T_1;L^2(\Omega ))}+\Vert \textbf{v}_\varepsilon -\textbf{v}_A\Vert _{L^\infty (0,T_1;L^2(\Omega ))\cap L^2(0,T_1;H^1(\Omega ))}&\le C_R\varepsilon ^{N+\frac{1}{4}}, \end{aligned}$$

(1.18)

where $\Gamma (\tilde{\delta })$ are standard tubular neighbourhoods for $\tilde{\delta }\in [0,3\delta ]$, $\delta >0$ small and $\nabla _{{\varvec{\tau }}_\varepsilon }$ is a suitable (approximate) tangential gradient, see Sect. 2.1. Moreover, let $d_\Gamma$ be the signed distance to $\Gamma$. Then

$$\begin{aligned} c_A&=\zeta (d_\Gamma )\theta _0(\rho _\varepsilon ) \pm \chi _{\Omega ^\pm } (1-\zeta (d_\Gamma )) + O(\varepsilon ^{\frac{3}{2}})&\quad&\text { in }L^\infty ((0,T_0)\times \Omega ),\end{aligned}$$

(1.19)

$$\begin{aligned} \textbf{v}_A&=\textbf{v}_0^+(x,t)\eta (\rho _\varepsilon )+\textbf{v}_0^-(x,t)(1-\eta (\rho _\varepsilon )) + O(\sqrt{\varepsilon })&\quad&\text { in }L^\infty (0,T_0;L^p(\Omega )), \end{aligned}$$

(1.20)

where $\varepsilon \rho _\varepsilon =d_\Gamma +O(\sqrt{\varepsilon })$ in $\Gamma (3\delta )$, $p\in [1,\infty )$ is arbitrary, $\zeta :\mathbb {R}\rightarrow [0,1]$ is smooth such that ${\text {supp}}\zeta \subseteq [-2\delta ,2\delta ]$ and $\zeta \equiv 1$ on $[-\delta ,\delta ]$, and $\eta :\mathbb {R}\rightarrow [0,1]$ is smooth such that $\eta =0$ in $(-\infty ,-1]$, $\eta =1$ in $[1,\infty )$, $\eta -\frac{1}{2}$ is odd and $\eta '\ge 0$ in $\mathbb {R}$. In particular,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} c_A=\pm 1\quad \text { uniformly on compact subsets of }\quad \Omega ^\pm . \end{aligned}$$

Remark 1.2

Note that for strong solutions of (1.1)–(1.5) we have the energy inequality

$$\begin{aligned} \sup _{t\in [0,T_0]}\int _\Omega \tfrac{1}{2}{|\textbf{v}_\varepsilon (t)|^2}+\tfrac{\varepsilon }{2}|\nabla c_\varepsilon (t)|^2 + \tfrac{1}{\varepsilon }f(c_\varepsilon (t))\, \textrm{d}x+ \int _0^{T_0}\int _\Omega |D\textbf{v}_\varepsilon |^2+ \frac{m_\varepsilon }{\varepsilon } |\mu _\varepsilon |^2 \, \textrm{d}x \, \textrm{d}t \le E_{0,\varepsilon }, \nonumber \\ \end{aligned}$$

(1.21)

where $\mu _\varepsilon = -\varepsilon \Delta c_\varepsilon + \tfrac{1}{\varepsilon } f'(c_\varepsilon )$ and

$$\begin{aligned} E_{0,\varepsilon }:= \int _\Omega \tfrac{1}{2}{|\textbf{v}_{0,\varepsilon }|^2} \,\textrm{d}x +\int _\Omega (\tfrac{\varepsilon }{2}|\nabla c_{0,\varepsilon }|^2 + \tfrac{1}{\varepsilon }f(c_{0,\varepsilon }))\, \textrm{d}x. \end{aligned}$$

Therefore the left-hand side of (1.21) is uniformly bounded in $\varepsilon \in (0,1)$ if $\sup _{\varepsilon \in (0,1)} E_{0,\varepsilon }<\infty$. Using a Taylor expansion for f and the form of $c_A$, $\textbf{v}_A$ in Sect. 4 below, one can show this bound under the assumption (1.14).

Let us comment on the novelty of our contribution. We use a similar strategy as in Abels and Fei [3]. Compared to [3], we consider the case of vanishing mobility $m_\varepsilon =\sqrt{\varepsilon }$ in (1.4), leading to the classical two-phase Navier–Stokes system with surface tension (1.6)–(1.12) in the sharp interface limit instead of the coupling with mean curvature flow in (1.11) obtained in [3]. Some remarks on the choice of the scaling for the mobility were included before. Note that our choice turns out to be critical for the arguments we use, and therefore we need to take time small in our result compared to [3]. Moreover, we need fractional order expansions with $\sqrt{\varepsilon }$-spacing in the terms, cf. Sect. 3 below. Additionally, note that in [3] a new type of ansatz in higher orders was introduced based on a linearization idea that simplified the previous works [5,6,7]. However, a direct modification with uncoupled equations for the higher order ansatz terms as in [3] does not lead to suitable estimates and hence is not enough to close the argument in our case. Therefore we modify this type of ansatz and obtain as model problem a coupled system (and another uncoupled problem in higher order) with suitable scaling in $\varepsilon$, see Sects. 2.3 and 4 below. Moreover, we even have a term at order $O(\sqrt{\varepsilon })$ in the expansion of the distance function which leads to problems when applying spectral estimates within standard tubular neighbourhood coordinates. Therefore we use the novel idea of working with $\varepsilon$-dependent coordinates, in particular as framework for the spectral estimates, cf. Sects. 2.1 and 2.4 below.

Finally, let us summarize the structure of the paper. Section 2 contains the required preliminaries, i.e., $\varepsilon$-dependent coordinates, estimates of remainder terms, the (coupled and uncoupled) model problems with scalings in $\varepsilon$ as well as spectral estimates based on the $\varepsilon$-scaled coordinates. The asymptotic expansion is done in Sect. 3, where the novelty lies in the expansion in integer powers of $\sqrt{\varepsilon }$ instead of integer powers of $\varepsilon$. The sophisticated higher order ansatz terms and remainder estimates are the content of Sect. 4. Finally, the main result is proven in Sect. 5, where a major part is the control of the error in the velocities in Sect. 5.1.

2 Preliminaries

Throughout the manuscript $\mathbb {N}$ denotes the set of natural numbers (without 0) and $\mathbb {N}_0=\mathbb {N}\cup \{0\}$. Let ${U}\subseteq \mathbb {R}^N$ be open, $m\in \mathbb {N}_0$, $p\in [1,\infty ]$ and X be a Banach space. Then we denote with $L^p({U};X)$ and $W^m_p({U};X)$ the standard Lebesgue and Sobolev spaces. In the case $X=\mathbb {R}$ we write $L^p({U})$ and $W^m_p({U})$, respectively. Moreover, if U has finite measure, we define for $1\le q\le \infty$ and $k\in \mathbb {N}_0$

$$\begin{aligned} L^q_{(0)}({U})&:= \left\{ f\in L^q({U}): \int _{U} f(x)\, \textrm{d}x=0\right\} , \quad W^k_{q,(0)}({U}) := W^k_q({U})\cap L^q_{(0)}({U}). \end{aligned}$$

Finally, note that we use the convention that $\nabla$, $div$ and $\Delta$ only act on spatial variables and not on rescaled ones.

2.1 Coordinates

Let $\Omega \subseteq \mathbb {R}^2$ be a domain, $T_0>0$ and $\Gamma =\bigcup _{t\in [0,T_0]}\Gamma _t\times \{t\}$ be a smooth evolving compact closed curve contained in $\Omega$. Then $\Omega$ is divided into two disjoint connected components $\Omega ^\pm _t$ such that $\partial \Omega ^+_t=\Gamma _t$ for all $t\in [0,T_0]$. We parametrize $\Gamma _t$ for every $t\in [0,T_0]$ over the torus $\mathbb {T}^1=\mathbb {R}/2\pi \mathbb {Z}$ with an $X_0:\mathbb {T}^1\times [0,T_0]\rightarrow \Gamma$ such that $\partial _sX_0(s,t)\ne 0$ for all $s\in \mathbb {T}^1$, $t\in [0,T_0]$. Moreover, we denote the corresponding tubular neighbourhood coordinates with $(d_0,S_0):\overline{\Gamma (3\delta )}\rightarrow [-3\delta ,3\delta ]\times \mathbb {T}^1$, where $\Gamma (\tilde{\delta }):=\{d_0\in (-\tilde{\delta },\tilde{\delta })\}$ is a relatively open neighbourhood of $\Gamma$ in $\Omega \times [0,T_0]$ for $\tilde{\delta }\in (0,3\delta ]$ and $\delta >0$ is small such that $\overline{\Gamma (3\delta )}\subseteq \Omega \times [0,T_0]$. Here $d_0(\cdot ,t)$ is the signed distance function to $\Gamma _t$ for every $t\in [0,T_0]$ and $d_0(x,t)\gtrless 0$ in $\Omega ^\pm _t$. We set $\Gamma _t(\tilde{\delta }):=\{x \in \Omega : (x,t)\in \Gamma (\tilde{\delta })\}$ for $\tilde{\delta }\in (0,3\delta ]$ and we also write $d_\Gamma :=d_0$. Moreover, we denote with

$$\begin{aligned} {\varvec{\tau }}(s,t):=\frac{\partial _{s} X_0(s,t)}{|\partial _s X_0(s,t)|}\quad \text {and}\quad \textbf{n}(s,t):= \begin{pmatrix} 0 &{} -1\\ 1 &{} 0 \end{pmatrix} {\varvec{\tau }}(s,t), \quad \text { for }(s,t)\in \mathbb {T}^1\times [0,T_0] \end{aligned}$$

the unit tangent and normal vectors of $\Gamma _t$ at $X_0(s,t)$, where $X_0$ is chosen such that $\textbf{n}(s,t)$ is the interior normal with respect to $\Omega ^+(t)$ for all $t\in [0,T_0]$. Moreover, we define

$$\begin{aligned} \textbf{n}_{\Gamma _t}(x):= \textbf{n}(s,t)\quad \text { for all }~ x=X_0(s,t)\in \Gamma _t, \end{aligned}$$

and let $V_{\Gamma _t}$ and $H_{\Gamma _t}$ be the normal velocity and (mean) curvature of $\Gamma _t$ with respect to $\textbf{n}_{\Gamma _t}$ for $t\in [0,T_0]$. We denote

$$\begin{aligned} V(s,t):=V_{\Gamma _t}|_{X_0(s,t)},\quad H(s,t):= H_{\Gamma _t}|_{X_0(s,t)}\quad \text {for all }(s,t)\in \mathbb {T}^1\times [0,T_0]. \end{aligned}$$

Here and in the following $u|_{X_0}:\mathbb {T}^1\times [0,T_0]\rightarrow \mathbb {R}$ is defined by

$$\begin{aligned} u|_{X_0}(s,t):= u|_{X_0(s,t)}:= u(X_0(s,t))\quad \text {for all }(s,t)\in \mathbb {T}^1\times [0,T_0] \end{aligned}$$

for a function u defined on a set containing $\Gamma$. It is well known that

$$\begin{aligned} { |\nabla d_\Gamma |^2}&{\equiv 1, \quad \nabla d_\Gamma \cdot \nabla S_0=0\quad \text { in }\Gamma (3\delta ),}\\ \nabla d_{\Gamma }|_{X_0}&=\textbf{n},~\quad \partial _t d_{\Gamma }|_{X_0}=-V,\quad \Delta d_{\Gamma }|_{X_0}=-H\quad \text { on }\mathbb {T}^1\times [0,T_0]. \end{aligned}$$

Later we will need a suitable $\varepsilon$-perturbation of the standard tubular neighbourhood coordinate system. Therefore we consider for $\eta >0$ and $\varepsilon \in (0,\varepsilon _0]$

$$\begin{aligned} d_\varepsilon (x,t)&:=d_0(x,t)+\varepsilon ^\eta \tilde{d}_\varepsilon (x,t),\end{aligned}$$

(2.1)

$$\begin{aligned} S_\varepsilon (x,t)&:=S_0(x,t)+\varepsilon ^\eta \tilde{S}_\varepsilon (x,t)/2\pi \quad \text { for }(x,t)\in \overline{\Gamma (3\delta )}, \end{aligned}$$

(2.2)

where $(\tilde{d}_\varepsilon ,\tilde{S}_\varepsilon ):\overline{\Gamma (3\delta )}\rightarrow \mathbb {R}^2$ are smooth with $C^k$-norm uniformly bounded with respect to $\varepsilon \in (0,\varepsilon _0]$ for every $k\in \mathbb {N}$, and we assume that

$$\begin{aligned} \tilde{d}_\varepsilon =0\quad \text { on }\quad \Gamma (3\delta )\backslash \Gamma (\delta ') \end{aligned}$$

(2.3)

for some $\delta '\in (0,3\delta )$. For small $\varepsilon$ these coordinates also have suitable properties similar to a tubular neighbourhood system because of the following theorem.

Theorem 2.1

($\varepsilon$-Coordinates) For $\varepsilon _1>0$ sufficiently small and every $\varepsilon \in (0,\varepsilon _1]$ the $\varepsilon$-coordinates $(d_\varepsilon ,S_\varepsilon ,id _t):\overline{\Gamma (3\delta )}\rightarrow [-3\delta ,3\delta ]\times \mathbb {T}^1\times [0,T_0]$ are well-defined and yield a smooth diffeomorphism with inverse $X_\varepsilon$. Moreover, for $\varepsilon _1$ small

$$\begin{aligned} \Gamma (\delta )\subseteq \Gamma ^\varepsilon (\tfrac{3\delta }{2}) \subseteq \Gamma (2\delta )\subseteq \Gamma ^\varepsilon (\tfrac{9\delta }{4}) \subseteq \Gamma (\tfrac{5\delta }{2})\subseteq \Gamma ^\varepsilon (\tfrac{11\delta }{4}) \subseteq \Gamma (3\delta ) \end{aligned}$$

(2.4)

for all $\varepsilon \in (0,\varepsilon _1]$, where for $\delta '>0$

$$\begin{aligned} \Gamma ^\varepsilon (\delta '):=\big \{(x,t)\in \Gamma (3\delta ): d_\varepsilon (x,t)\in (-\delta ',\delta ')\big \}. \end{aligned}$$

Moreover, for every $k\in \mathbb {N}$ the $C^k$-norms of $d_\varepsilon , S_\varepsilon$ are uniformly bounded.

Proof

Because of (2.3) we obtain that $(d_\varepsilon ,S_\varepsilon ,id _t):\overline{\Gamma (3\delta )}\rightarrow [-3\delta ,3\delta ]\times \mathbb {T}^1\times [0,T_0]$ is well-defined and smooth for $\varepsilon >0$ small. Moreover, because of compactness and the definitions it holds that $D(d_\varepsilon ,S_\varepsilon ,id _t)$ is invertible pointwise in $\overline{\Gamma (3\delta )}$ for $\varepsilon >0$ small. Hence $(d_\varepsilon ,S_\varepsilon ,id _t)$ is a local diffeomorphism. Furthermore, a compactness and extension argument shows that $(d_0,S_0,id _t)$ is globally bi-Lipschitz. This extends to $(d_\varepsilon ,S_\varepsilon ,id _t)$ with uniform constants for $\varepsilon >0$ small. Injectivity of $(d_\varepsilon ,S_\varepsilon ,id _t)$ directly follows and surjectivity can now be proven by showing that the image is open and closed in the connected space $[-3\delta ,3\delta ]\times \mathbb {T}^1$. The additional statement is clear from the definitions for $\varepsilon >0$ small. $\square$

As before we define

$$\begin{aligned} \Gamma ^\varepsilon _t(\tilde{\delta }):=\{x \in \Omega : (x,t)\in \Gamma ^\varepsilon (\tilde{\delta })\}\quad \text {for }\tilde{\delta }\in (0,3\delta ]. \end{aligned}$$

Remark 2.2

In order to transform integrals with $X_\varepsilon$ later, we define $J_\varepsilon :[-3\delta ,3\delta ]\times \mathbb {T}^1\times [0,T_0]\rightarrow (0,\infty )$ by

$$\begin{aligned} J_\varepsilon :=|\det DX_\varepsilon | = \frac{1}{\sqrt{|\nabla d_\varepsilon |^2|\nabla S_\varepsilon |^2-(\nabla d_\varepsilon \cdot \nabla S_\varepsilon )^2}} \circ X_\varepsilon . \end{aligned}$$

(2.5)

Furthermore, we denote

$$\begin{aligned} \textbf{n}_\varepsilon (x,t):= \nabla d_\varepsilon (x,t)\qquad \text {for all }(x,t)\in \overline{\Gamma (3\delta )}. \end{aligned}$$

For the following we assume that

$$\begin{aligned} |\textbf{n}_\varepsilon |^2=|\nabla d_\varepsilon |^2=1+O(\varepsilon ^2),\quad \textbf{n}_\varepsilon \cdot \nabla S_\varepsilon = O(\varepsilon ^2) \end{aligned}$$

(2.6)

in $C^k_b(\overline{\Gamma (3\delta )})$ for all $k\in \mathbb {N}$, which we will assure in the following constuction. The following identity will be useful in relation with divergence free functions:

$$\begin{aligned} \partial _r X_\varepsilon (r,s,t) \otimes \textbf{n}_\varepsilon (X_\varepsilon (r,s,t),t)+ \partial _s X_\varepsilon (r,s,t)\otimes \nabla S_\varepsilon (X_\varepsilon (r,s,t),t) = I \end{aligned}$$

(2.7)

for all $r\in (-3\delta ,3\delta )$, $s\in \mathbb {T}^1, t\in [0,T_0]$. It is a consequence of differentiating

$$\begin{aligned} X_\varepsilon (d_\varepsilon (x,t), S_\varepsilon (x,t))= x\qquad \text {for all }(x,t)\in \Gamma ^\varepsilon (3\delta ). \end{aligned}$$

This motivates to define for suitable $\psi$

$$\begin{aligned} \nabla _{{\varvec{\tau }}_\varepsilon } \psi := \nabla S_\varepsilon [\partial _s (\psi \circ X_\varepsilon )]{\circ X_\varepsilon ^{-1}}. \end{aligned}$$

(2.8)

Then

$$\begin{aligned} \nabla \psi = \nabla _{{\varvec{\tau }}_\varepsilon } \psi + \textbf{n}_\varepsilon (\partial _r X_\varepsilon \circ X_\varepsilon ^{-1}\cdot \nabla ) \psi . \end{aligned}$$

Moreover, (2.7) implies

$$\begin{aligned} \partial _r X_\varepsilon (r,s,t)&= \textbf{n}_\varepsilon (X_\varepsilon (r,s,t),t)+O(\varepsilon ^2)= \textbf{n}_\varepsilon (X_\varepsilon (0,s,t),t)+O(\varepsilon ^2) \end{aligned}$$

(2.9)

in $C^k_b(\overline{\Gamma (3\delta )})$ for all $k\in \mathbb {N}$ due to (2.6) and

$$\begin{aligned} \partial _r \textbf{n}_\varepsilon (X_\varepsilon (r,s,t))= \nabla \textbf{n}_\varepsilon (X_\varepsilon (r,s,t))\cdot \partial _r X_\varepsilon (r,s,t)= O(\varepsilon ^2) \end{aligned}$$

in $C^k_b(\overline{\Gamma (3\delta )})$ for all $k\in \mathbb {N}$. In particular this shows

$$\begin{aligned} (\partial _r X_\varepsilon )\circ X_\varepsilon ^{-1}\cdot \nabla u = \partial _{\textbf{n}_\varepsilon } u+ O(\varepsilon ^2) \end{aligned}$$

for every sufficiently smooth $u :\Gamma (3\delta )\rightarrow \mathbb {R}$, where $\partial _{\textbf{n}_\varepsilon } u:= \textbf{n}_\varepsilon \cdot \nabla u$. Similarly, by multiplying (2.7) with $\nabla S_\varepsilon (X_\varepsilon (r,s,t),t)$ one obtains

$$\begin{aligned} \partial _s X_\varepsilon (r,s,t)= |\nabla S(X_\varepsilon (r,s,t))|^{-2} \nabla S(X_\varepsilon (r,s,t)+ O(\varepsilon ^2) \end{aligned}$$

in $C^k_b(\overline{\Gamma (3\delta )})$ for all $k\in \mathbb {N}$.

Remark 2.3

Let $a:\Gamma (3\delta )\rightarrow \mathbb {R}$ be smooth in normal direction and assume $a=0$ on $\Gamma$, then $\tilde{a}:\Gamma (3\delta )\rightarrow \mathbb {R}$ with

$$\begin{aligned} \tilde{a}(x,t):= {\left\{ \begin{array}{ll} \frac{a(x,t)}{d_\Gamma (x,t)} &{}\text { for all }(x,t)\in \Gamma (3\delta )\setminus \Gamma ,\\ \partial _{\textbf{n}} a(x,t) &{}\text { for all }(x,t)\in \Gamma \end{array}\right. } \end{aligned}$$

is well-defined, smooth in normal direction and tangential regularity is conserved. In particular $\tilde{a}$ is smooth provided that a is smooth. This can be shown with a Taylor expansion in $d_\Gamma$.

A similar statement, based on a Taylor expansion in normal direction for Sobolev functions, is given by the following lemma and will be useful to estimate remainder terms.

Lemma 2.4

Let $t\in [0,T_0]$, $\varepsilon \in (0,\varepsilon _1)$ with $\varepsilon _1>0$ as in Theorem 2.1, $\delta '\in [2\delta ,3\delta ]$ and $a\in W^k_p(\Gamma _t(\delta '))$ for some $k\in \mathbb {N}$, $1< p< \infty$. Then there are $r_{k,\varepsilon ,t} \in L^p(\Gamma _t(\delta '))$ such that

$$\begin{aligned} a(x)= \sum _{j=0}^{k-1} (\partial _{\textbf{n}_\varepsilon }^j a)(P_\varepsilon (x,t))\frac{d_\varepsilon (x,t)^j}{j!} + d_\varepsilon (x,t)^k r_{k,\varepsilon ,t}(x)\quad \text {for all }x\in \Gamma _t(\delta '), \end{aligned}$$

where $P_\varepsilon (x,t):= X_\varepsilon (0,S_\varepsilon (x,t),t)$, $\textbf{n}_\varepsilon (x,t)= \nabla d_\varepsilon (x,t)$, and

$$\begin{aligned} \Vert r_{k,\varepsilon ,t}\Vert _{L^p(\Gamma _t(\delta '))}\le C_k \Vert a\Vert _{W^k_p(\Gamma _t(\delta '))}. \end{aligned}$$

(2.10)

for some $C_k$ independent of $\varepsilon ,t$, and a.

Proof

Since smooth functions are dense in $W^k_p(\Gamma _t(\delta '))$, we can assume that a is smooth. We define the auxiliary function $\Phi _{x,t}(r):= X_\varepsilon (r,S_\varepsilon (x,t),t)$ for all $r\in [-\delta ',\delta ']$. Then by a one-dimensional Taylor expansion

$$\begin{aligned} a(x)&= a(\Phi _{x,t}(d_\varepsilon (x,t))\\&= \sum _{j=0}^{k-1} \frac{d^j}{dr^j} (a(\Phi _{x,t}(r)))|_{r=0} \frac{d_\varepsilon (x,t)^j}{j!} + \int _0^{d_\varepsilon (x,t)}\frac{d^k(a(\Phi _{x,t}(r)))}{dr^k} \frac{(d_\varepsilon (x,t)-r)^{k-1}}{(k-1)!} \,\textrm{d}r\\&= \sum _{j=0}^{k-1} (\partial _{\textbf{n}_\varepsilon }^j a)(P_\varepsilon (x,t))\frac{d_\varepsilon (x,t)^j}{j!} + d_\varepsilon (x,t)^kr_{k,\varepsilon ,t}(x) \end{aligned}$$

where

$$\begin{aligned} r_{k,\varepsilon ,t}(x)= \frac{1}{d_\varepsilon (x,t)}\int _0^{d_\varepsilon (x,t)}\frac{d^k(a(\Phi _x(r)))}{dr^k} \frac{(1-r/d_\varepsilon (x,t))^{k-1}}{(k-1)!} \,\textrm{d}r. \end{aligned}$$

Now using that by Hardy’s inequality

$$\begin{aligned} \left( \int _{-\delta '}^{\delta '}\left| \frac{1}{\tilde{r}} \int _0^{\tilde{r}}\left| \frac{d^k(a(\Phi _{x,t}(r)))}{dr^k}\right| \,\textrm{d}r \right| ^p\, \textrm{d}\tilde{r} \right) ^{\frac{1}{p}}\le C_p \left( \int _{-\delta '}^{\delta '}\left| \frac{d^k(a(\Phi _{x,t}(r)))}{dr^k}\right| ^p\,\textrm{d}r \right) ^{\frac{1}{p}} \end{aligned}$$

one easily shows (2.10). $\square$

For the following let $h:\mathbb {T}^1\times [0,T_0]\rightarrow \mathbb {R}$ be sufficiently smooth. Then we have

$$\begin{aligned} \begin{aligned} \nabla ^{\Gamma _t^\varepsilon } h(r,s,t)&:= \nabla _x (h(S_\varepsilon (x,t),t) = \nabla S_\varepsilon (x,t) \partial _{s} h(s,t),\\ \Delta ^{\Gamma _t^\varepsilon } h(r,s,t)&:= \Delta _x (h(S_\varepsilon (x,t),t) = (\Delta S_\varepsilon )(x,t)\cdot \partial _s h(s,t)+|\nabla S_\varepsilon (x,t)|^2 \partial _{s}^2 h(s,t) \end{aligned}\nonumber \\ \end{aligned}$$

(2.11)

for all $(x,t)\in \Gamma (3\delta )$, where $r\in (-3\delta ,3\delta )$ and $s\in \mathbb {T}^1$ are determined by $x=X_\varepsilon (r,s,t)$. Therefore we define for every sufficiently smoth $h:\mathbb {T}^1\times [0,T_0]\rightarrow \mathbb {R}$

$$\begin{aligned} \begin{aligned} (\nabla _{\Gamma _t^\varepsilon } h)(s,t)&:=(\nabla ^{\Gamma _t^\varepsilon } h)(0,s,t),\quad (\Delta _{\Gamma _t^\varepsilon } h)(s,t):=(\Delta ^{\Gamma _t^\varepsilon } h)(0,s,t)\quad \text {for all }s\in \mathbb {T}^1,t\in [0,T_0]. \end{aligned} \end{aligned}$$

We note that coefficients of the differences

$$\begin{aligned} (\nabla ^{\Gamma _t^\varepsilon } h)(r,s,t)-(\nabla _{\Gamma _t^\varepsilon } h)(s,t),\quad (\Delta ^{\Gamma _t^\varepsilon } h)(r,s,t)-(\Delta _{\Gamma _t^\varepsilon } h)(s,t) \end{aligned}$$

vanish for $r=0$, which corresponds to $x\in \Gamma _t^\varepsilon$.

Finally, let $U_t\subseteq \mathbb {R}^2$, $t\in [0,T]$, be open sets and $\mathcal {U}:=\bigcup _{t\in [0,T]} U_t$. Then we define for $s\ge 0$

$$\begin{aligned} L^2(0,T;H^s(U_t))&:= \left\{ g\in L^2(\mathcal {U}): g(\cdot ,t)\in H^s(U_t)\text { for a.e.\ }t\in (0,T), \Vert g(\cdot ,t)\Vert _{H^s(U_t)}\in L^2(0,T)\right\} ,\\ \Vert g\Vert _{L^2(0,T;H^s(U_t))}&:= \left( \int _0^T \Vert g(\cdot ,t)\Vert _{H^s(U_t)}^2\, \textrm{d}t\right) ^{\frac{1}{2}}, \\ L^2(0,T;H^s(\Gamma _t))&:= \left\{ g\in L^2(\Gamma ): g\circ X_0 \in L^2(0,T;H^s(\mathbb {T}^1)) \right\} ,\\ \Vert g\Vert _{L^2(0,T;H^s(\Gamma _t))}&:= \Vert g\circ X_0\Vert _{L^2(0,T;H^s(\mathbb {T}^1))}. \end{aligned}$$

2.2 The stretched variable and remainder terms

In the following we will use a “stretched variable”, which is defined by

$$\begin{aligned} \rho =\rho _\varepsilon := \frac{d_\varepsilon (x,t)}{\varepsilon }\qquad \text {for } (x,t)\in \Gamma (3\delta ), \varepsilon \in (0,\varepsilon _1], \end{aligned}$$

(2.12)

where $d_\varepsilon :\Gamma (3\delta )\rightarrow \mathbb {R}$ is as in the previous subsection and $\varepsilon _1>0$ is as in Theorem 2.1. In particular, it satisfies

$$\begin{aligned} |\nabla d_\varepsilon (x,t)| = 1+ \varepsilon ^2b_\varepsilon (x,t) \qquad \text {for all }(x,t)\in \Gamma (3\delta ), \end{aligned}$$

where $b_\varepsilon$ and all its derivatives are uniformly bounded in $\varepsilon \in (0,\varepsilon _1]$ for some $\varepsilon _1>0$ sufficiently small.

For a systematic treatment of the remainder terms, we introduce:

Definition 2.5

For any $k\in \mathbb {R}$ and $\alpha >0$, $\mathcal {R}_{k,\alpha }$ denotes the vector space of family of continuous functions $\hat{r}_\varepsilon :\mathbb {R}\times \Gamma (3\delta ) \rightarrow \mathbb {R}$, indexed by $\varepsilon \in (0,1)$, which are continuously differentiable with respect to $\textbf{n}_{\Gamma _t}$ for all $t\in [0,T_0]$ such that

$$\begin{aligned} |\partial _{\textbf{n}_{\Gamma _t}}^j \hat{r}_\varepsilon (\rho ,x,t)|\le Ce^{-\alpha |\rho |}\varepsilon ^k \,\,\, \text {for all }\rho \in \mathbb {R},(x,t)\in \Gamma (3\delta ), j=0,1, \varepsilon \in (0,1)\qquad \end{aligned}$$

(2.13)

for some $C>0$ independent of $\rho \in \mathbb {R},(x,t)\in \Gamma (3\delta )$, $\varepsilon \in (0,1)$. $\mathcal {R}_{k,\alpha }^0$ is the subclass of all $(\hat{r}_\varepsilon )_{\varepsilon \in (0,1)}\in \mathcal {R}_{k,\alpha }$ such that $\hat{r}_\varepsilon (\rho ,x,t)= 0$ for all $\rho \in \mathbb {R}, x\in \Gamma _t, t\in [0,T_0]$.

We remark that $\mathcal {R}_{k,\alpha }$ and $\mathcal {R}^0_{k,\alpha }$ are closed under multiplication and $\mathcal {R}_{k,\alpha }\subset \mathcal {R}_{k-1,\alpha }$.

Lemma 2.6

Let $0<\varepsilon \le \varepsilon _1$, $d_\varepsilon$ be defined as before, $\delta '\in [\tfrac{\delta }{2}, 3\delta ]$ and $(\hat{r}_\varepsilon )_{0<\varepsilon <1}\in L^2(\mathbb {T}^1;L^1(\mathbb {R})\cap L^2(\mathbb {R}))$, $(a_\varepsilon )_{\varepsilon \in (0,1)} \subseteq C(\overline{\Gamma (3\delta )})$ such that

$$\begin{aligned} |a_\varepsilon (x,t)|\le C|d_\varepsilon (x,t)|^j\qquad \text {for all }(x,t)\in \Gamma (3\delta ), \varepsilon \in (0,\varepsilon _1] \end{aligned}$$

for some $C>0$ independent of $\varepsilon ,x,t$ and $j\in \mathbb {N}_0$. Then there is some $C>0$, independent of $0<\varepsilon \le \varepsilon _1$, $\varepsilon _1\in (0,1)$ such that for each $t\in [0,T]$

$$\begin{aligned} r_\varepsilon (x):= a_\varepsilon (x,t) \hat{r}_\varepsilon \left( \frac{d_\varepsilon (x,t)}{\varepsilon }, S_\varepsilon (x,t)\right) \qquad \text {for }x\in \Gamma _t(3\delta ) \end{aligned}$$

satisfies

$$\begin{aligned} \left\| r_\varepsilon \varphi \right\| _{L^1(\Gamma _t^\varepsilon (\delta '))}&\le C\varepsilon ^{1+j}\Vert \rho ^j\hat{r}_\varepsilon \Vert _{L^2(\mathbb {T}^1;L^1(\mathbb {R}))}\Vert \varphi \Vert _{H^1(\Gamma _t^\varepsilon (\delta '))},\end{aligned}$$

(2.14)

$$\begin{aligned} \left\| r_\varepsilon \right\| _{L^2(\Gamma _t^\varepsilon (\delta '))}&\le C\varepsilon ^{\frac{1}{2}+j} \Vert \rho ^j\hat{r}_\varepsilon \Vert _{L^2(\mathbb {T}^1;L^2(\mathbb {R}))} \end{aligned}$$

(2.15)

uniformly for all $\varphi \in H^1(\Gamma _t^\varepsilon (\delta '))$, $t\in [0,T_0]$, and $\varepsilon \in (0,\varepsilon _1]$.

Proof

With the aid of the change of variables $x= X_\varepsilon (r,s)$, where $r=d_\varepsilon (x,t)$, $s=S_\varepsilon (x,t)$, we obtain

$$\begin{aligned} \left\| r_\varepsilon \varphi \right\| _{L^1(\Gamma _t^\varepsilon (\delta '))}&=\int _{-\delta '}^{\delta '}\int _{\mathbb {T}^1} |a_\varepsilon (X_\varepsilon (r,s,t),t)| \left| \hat{r}_\varepsilon \left( \tfrac{r}{\varepsilon },s,t \right) \right| |\varphi (X_\varepsilon (r,s,t))|J_\varepsilon (r,s,t)\,\textrm{d}s\,\textrm{d}r \\&\le C\int _{\mathbb {T}^1}\int _\mathbb {R}|r^j|| \hat{r}_\varepsilon (\tfrac{r}{\varepsilon }, s,t)|\,\textrm{d}r \sup _{r\in (-\delta ',\delta ')}|\varphi (X_\varepsilon (r,s,t))| \,\textrm{d}s\\&\le C \varepsilon ^{1+j}\left( \int _{\mathbb {T}^1}\left| \int _{\mathbb {R}} |\rho ^j \hat{r}_\varepsilon (\rho ,s,t)|\,\textrm{d}\rho \right| ^2 \textrm{d}s\right) ^{\frac{1}{2}} \left( \int _{\mathbb {T}^1}\sup _{|r|\le \delta ' }|\varphi (X_\varepsilon (r,s,t))|^2\,\textrm{d}s\right) ^{\frac{1}{2}}\\&\le C\varepsilon ^{1+j}\Vert \rho ^j\hat{r}_\varepsilon \Vert _{L^2(\mathbb {T}^1;L^1(\mathbb {R}))} \Vert \varphi \Vert _{H^1(\Gamma ^\varepsilon _t(\delta '))} \end{aligned}$$

for all $\varepsilon \in (0,\varepsilon _1]$, $t\in [0,T_0]$, and $\varphi \in H^1(\Omega )$. This proves the first estimate.

In the same way we estimate

$$\begin{aligned}&\left\| r_\varepsilon \right\| _{L^2(\Gamma _t^\varepsilon (\frac{3\delta }{2}))}^2 =\int _{-\delta '}^{\delta '}\int _{\mathbb {T}^1} |a_\varepsilon (X_\varepsilon (r,s,t),t)|^2 \left| \hat{r}_\varepsilon \left( \tfrac{r}{\varepsilon },s,t \right) \right| ^2J_\varepsilon (r,s,t)\,\textrm{d}s\,\textrm{d}r \\&\le C\int _{\mathbb {T}^1}\int _\mathbb {R}|r^j|^2| \hat{r}_\varepsilon (\tfrac{r}{\varepsilon }, s,t)|^2\,\textrm{d}r\,\textrm{d}s= C\varepsilon ^{1+2j}\Vert \rho ^j\hat{r}_\varepsilon \Vert _{L^2(\mathbb {T}^1\times \mathbb {R})}^2 \end{aligned}$$

for all $\varepsilon \in (0,\varepsilon _1]$ and $t\in [0,T_0]$, which shows the second estimate. $\square$

Lemma 2.7

Let $g\in \mathcal {S}(\mathbb {R})$ and $\zeta \in C^\infty _0(\mathbb {R})$ with ${\text {supp}}\zeta \subseteq [-\tfrac{5\delta }{2},\tfrac{5\delta }{2}]$. Then there is constant $C>0$ such that for all $t\in [0,T_0]$, $a\in H^1(\mathbb {T}^1)$ and $\varvec{\varphi }\in H^1(\Omega )^d\cap L^2_\sigma (\Omega )$ we have

$$\begin{aligned} \left| \int _{\Gamma _t(3\delta )}\zeta \circ d_\Gamma g'(\rho _\varepsilon (x,t))a(S_\varepsilon (x,t))\textbf{n}_{\varepsilon }\otimes \textbf{n}_{\varepsilon }: \nabla \varvec{\varphi }(x) \, \textrm{d}x\right| \le C \varepsilon ^{\frac{3}{2}} \Vert a\Vert _{H^1(\mathbb {T}^1)}\Vert \varvec{\varphi }\Vert _{H^1(3\delta )}, \end{aligned}$$

where $\textbf{n}_\varepsilon = \nabla d_\varepsilon$.

Proof

We use that

$$\begin{aligned} -\frac{\textbf{n}_{\varepsilon }}{|\textbf{n}_\varepsilon |}\otimes \frac{\textbf{n}_{\varepsilon }}{|\textbf{n}_\varepsilon |}: \nabla \varvec{\varphi }= \underbrace{\left( \textbf{I}-\frac{\textbf{n}_{\varepsilon }}{|\textbf{n}_\varepsilon |}\otimes \frac{\textbf{n}_{\varepsilon }}{|\textbf{n}_\varepsilon |}\right) }_{\textbf{P}_\varepsilon :=}: \nabla \varvec{\varphi } \end{aligned}$$

(2.16)

since ${\text {div}}\varvec{\varphi }=0$. This together with $\varepsilon \textbf{n}_\varepsilon \cdot \nabla g(\rho _\varepsilon (x,t))= g'(\rho _\varepsilon (x,t))|\textbf{n}_\varepsilon |^2$ yields

$$\begin{aligned} I&:=\left| \int _{\Gamma _t(3\delta )} \zeta \circ d_\Gamma g'(\rho _\varepsilon (x,t))a(S_\varepsilon (x,t))\textbf{n}_{\varepsilon }\otimes \textbf{n}_{\varepsilon } : \nabla \varvec{\varphi } \, \textrm{d}x\right| \\&=\left| \int _{\Gamma _t(\frac{5\delta }{2})}\zeta \circ d_\Gamma \varepsilon (\textbf{n}_\varepsilon \cdot \nabla g(\rho _\varepsilon (x,t)))a(S_\varepsilon (x,t))\frac{\textbf{n}_{\varepsilon }}{|\textbf{n}_\varepsilon |}\otimes \frac{\textbf{n}_{\varepsilon }}{|\textbf{n}_\varepsilon |} : \nabla \varvec{\varphi } \, \textrm{d}x\right| \\&\le \left| \int _{\Gamma _t(\frac{5\delta }{2})} \varepsilon g(\rho _\varepsilon (x,t))\textbf{n}_\varepsilon \cdot \nabla \left( \zeta \circ d_\Gamma a(S_\varepsilon (x,t))\textbf{P}_\varepsilon : \nabla \varvec{\varphi }\right) \, \textrm{d}x\right| \\&\quad + \left| \int _{\Gamma _t(\frac{5\delta }{2})} \varepsilon g(\rho _\varepsilon (x,t)) {\text {div}}\textbf{n}_\varepsilon \left( \zeta \circ d_\Gamma a(S_\varepsilon (x,t))\textbf{P}_\varepsilon : \nabla \varvec{\varphi }\right) \, \textrm{d}x\right| \end{aligned}$$

for all $\varepsilon \in (0,\varepsilon _1)$, $t\in [0,T_0]$, where we used integration by parts in the last step. Using product rule we obtain

$$\begin{aligned} I&\le \left| \int _{\Gamma _t(\frac{5\delta }{2})} \varepsilon g(\rho _\varepsilon (x,t))\textbf{P}_\varepsilon : \nabla \left[ \zeta \circ d_\Gamma a(S_\varepsilon (x,t)) (\textbf{n}_\varepsilon \cdot \nabla ) \varvec{\varphi }\right] \, \textrm{d}x\right| \\&\quad +\left| \int _{\Gamma _t(\frac{5\delta }{2})} \varepsilon g(\rho _\varepsilon (x,t))(a(S_\varepsilon (x,t))\textbf{Q}_\varepsilon +(\partial _s a)(S_\varepsilon (x,t))\textbf{R}_\varepsilon ) :\nabla \varvec{\varphi } \, \textrm{d}x\right| \end{aligned}$$

for some uniformly bounded $\textbf{Q}_\varepsilon , \textbf{R}_\varepsilon$. Hence another integration by parts leads to

$$\begin{aligned} I&\le \left| \int _{\Gamma _t(\frac{5\delta }{2})} \zeta \circ d_\Gamma \varepsilon \textbf{P}_\varepsilon (\nabla g(\rho _\varepsilon (x,t)))a(S_\varepsilon (x,t))\cdot (\textbf{n}_\varepsilon \cdot \nabla ) \varvec{\varphi } \, \textrm{d}x\right| \\&\quad +\varepsilon C\int _{\Gamma _t^\varepsilon (\tfrac{11\delta }{4})} |g(\rho _\varepsilon (x,t))||a(S_\varepsilon (x,t))|+|(\partial _s a)(S_\varepsilon (x,t))||\nabla \varvec{\varphi }| \, \textrm{d}x, \end{aligned}$$

where

$$\begin{aligned} \textbf{P}_\varepsilon (\nabla g(\rho _\varepsilon (x,t)))= \textbf{P}_\varepsilon \left( \textbf{n}_\varepsilon \frac{1}{\varepsilon }g'(\rho _\varepsilon (x,t)\right) =0. \end{aligned}$$

Now using $g\in \mathcal {S}(\mathbb {R})$ and (2.15) we obtain

$$\begin{aligned} I&\le C\varepsilon ^{\frac{3}{2}}\left\| a\right\| _{H^{1}(\mathbb {T}^{1})}\left\| \varvec{\varphi }\right\| _{H^{1}\left( \Gamma _{t}(2\delta )\right) }. \end{aligned}$$

This finishes the proof. $\square$

2.3 Parabolic equations on evolving hypersurfaces

For $T\in (0,\infty )$ and $r\in [0,1]$ we shall denote the function spaces

$$\begin{aligned} X_{T,r}:=L^2(0,T;H^{2+r}(\mathbb {T}^1))\cap H^1(0,T;H^{r}(\mathbb {T}^1)),\quad X_T:=X_{T,\frac{1}{2}}. \end{aligned}$$

We equip $X_T$ with the norm

$$\begin{aligned} \Vert u\Vert _{X_T} =\Vert u\Vert _{L^2(0,T;H^{5/2}(\mathbb {T}^1))}+\Vert u\Vert _{H^1(0,T;H^{1/2}(\mathbb {T}^1))}+ \Vert u|_{t=0}\Vert _{H^{3/2}(\mathbb {T}^1)}. \end{aligned}$$

Then it holds

$$\begin{aligned} X_T\hookrightarrow C([0,T]; H^{3/2}(\mathbb {T}^1))\cap L^4(0,T; H^2(\mathbb {T}^1)) \end{aligned}$$

(2.17)

and the operator norm of the embedding is uniformly bounded in T.

In the following theorem we derive uniform estimates for a class of degenerate parabolic partial differential equations.

Theorem 2.8

Let $0<T\le T_1 <\infty$ and $\kappa \in (0,1]$, $r\in [0,1]$ and $a_\kappa ,b_\kappa ,c_\kappa :\mathbb {T}^1\times [0,T]\rightarrow \mathbb {R}$ be twice continuously differentiable with uniformly bounded $C^2$-norms with respect to $\kappa \in (0,1]$. Moreover, let there be some $c_0>0$, independent of $\kappa$, such that $c_\kappa (s,t) \ge c_0$ for all $(s,t)\in \mathbb {T}^1\times [0,T]$. For every $g\in L^2(0,T;H^r(\mathbb {T}^1))$ and $h_0\in H^{1+r}(\mathbb {T}^1)$ there is a unique solution $h\in X_{T,r}$ of

$$\begin{aligned} \partial _t h+ a_\kappa \partial _s h + b_\kappa h -\kappa c_\kappa \partial _s^2 h&= g&\qquad&\text {on }\mathbb {T}^1\times [0,T],\end{aligned}$$

(2.18)

$$\begin{aligned} h|_{t=0}&=h_0{} & {} \text {on } \mathbb {T}^1. \end{aligned}$$

(2.19)

Moreover, there is some $C=C(T_1)>0$ independent of $\kappa \in (0,1]$, $T\in (0,T_1]$, $h, g_\kappa , h_0$ such that

$$\begin{aligned} \Vert h\Vert _{C([0,T]; H^{r}(\mathbb {T}^1))} + \sqrt{\kappa }\Vert h\Vert _{L^2(0,T; H^{1+r}(\mathbb {T}^1))}&\le C\left( \Vert g\Vert _{L^1(0,T; H^{r}(\mathbb {T}^1))}+\Vert h_0\Vert _{H^{r}(\mathbb {T}^1)}\right) \end{aligned}$$

(2.20)

$$\begin{aligned} \sqrt{\kappa } \Vert h\Vert _{C([0,T]; H^{1+r}(\mathbb {T}^1))} + \kappa \Vert h\Vert _{L^2(0,T; H^{2+r}(\mathbb {T}^1))}&\le C\left( \Vert g\Vert _{L^2(0,T; H^r(\mathbb {T}^1))}+\Vert h_0\Vert _{H^{1+r}(\mathbb {T}^1)}\right) . \end{aligned}$$

(2.21)

Remark 2.9

Note that Theorem 2.8 can be applied for right hand sides $g_\kappa$ depending on $\kappa$.

Proof of Theorem 2.8

Existence of a unique solution follows by standard results on linear parabolic equations. Therefore we only need to prove the uniform estimates.

First we consider the case $r=0$. Then testing (2.18) with h and integrating with respect to t we obtain

$$\begin{aligned}&\sup _{0\le t\le T} \Vert h(t)\Vert _{L^2(\mathbb {T}^1)}^2 + \frac{\kappa c_0}{2} \int _0^T \int _{\mathbb {T}^1}|\partial _s h|^2\, \textrm{d}s\, \textrm{d}t \\&\quad \le C\sup _{s\in \mathbb {T}^1, t\in [0,T]} \left( |\partial _s a_\kappa (s,t)|+|b_\kappa (s,t)|+ |\partial _sc_\kappa (s,t)|^2 \right) \int _0^T \int _{\mathbb {T}^1}|h|^2\,\textrm{d}s \, \textrm{d}t\\&\qquad + \int _0^T \Vert g(t)\Vert _{L^2(\mathbb {T}^1)}\,\textrm{d}t \sup _{0\le t\le T} \Vert h(t)\Vert _{L^2(\mathbb {T}^1)}. \end{aligned}$$

Hence Young’s and Gronwall’s inequality imply (2.20).

Next let $r=1$. Then differentiating (2.18) with respect to s yields for $\tilde{h}=\partial _s h$

$$\begin{aligned} \partial _t \tilde{h}+ \tilde{a}_\kappa \partial _s \tilde{h} + \tilde{b}_\kappa \tilde{h} -\kappa c_\kappa \partial _s^2 \tilde{h} = \partial _s g\qquad \text {on }\mathbb {T}^1\times [0,T], \end{aligned}$$

(2.22)

for some $\tilde{a}_\kappa , \tilde{b}_\kappa :\mathbb {T}^1\times [0,T]\rightarrow \mathbb {R}$, which are smooth and have $C^1$-norms uniformly bounded in $\kappa \in (0,1]$. Hence the same estimate as before yields

$$\begin{aligned} \Vert \partial _sh\Vert _{C([0,T]; L^2(\mathbb {T}^1))} + \sqrt{\kappa }\Vert \partial _s h\Vert _{L^2(0,T; H^1(\mathbb {T}^1))} \le C\left( \Vert g\Vert _{L^1(0,T; H^{1}(\mathbb {T}^1))}+\Vert h_0\Vert _{H^{1}(\mathbb {T}^1)}\right) \end{aligned}$$

for some $C>0$ independent of $\kappa \in (0,1]$, $T\in (0,T_1]$ and $g,h_0$. This implies (2.20) in the case $r=1$. Finally, (2.20) for the case $r\in [0,1]$ follows by interpolation.

In order to prove (2.21) in the case $r=0$ we test (2.18) with $-\kappa \partial _s^2 h$ and obtain

$$\begin{aligned}&\kappa \frac{d}{dt} \int _{\mathbb {T}^1} \frac{|\partial _s h|^2}{2}\, \textrm{d}s + \kappa ^2 \int _{\mathbb {T}^1} c_\kappa |\partial _s^2h|^2\,\textrm{d}s = \kappa \int _{\mathbb {T}^1} a_\kappa \partial _s h \partial _s^2 h \, \textrm{d}s -\int _{\mathbb {T}^1} (g+b_\kappa h) \kappa \partial _s^2 h\,\textrm{d}s, \end{aligned}$$

where

$$\begin{aligned}&\kappa \int _{\mathbb {T}^1} a_\kappa \partial _s h \partial _s^2 h \, \textrm{d}s = -\kappa \int _{\mathbb {T}^1} \partial _s a_\kappa \frac{|\partial _s h|^2}{2}\,\textrm{d}s \le C\kappa \Vert h(t)\Vert _{H^1(\mathbb {T}^1)}^2,\\&\quad -\int _{\mathbb {T}^1} (g+b_\kappa h) \kappa \partial _s^2 h\,\textrm{d}s\le C(\Vert g(t)\Vert _{L^2(\mathbb {T}^1)}+\Vert h(t)\Vert _{L^2(\mathbb {T}^1)})\kappa \Vert \partial _s^2 h(t)\Vert _{L^2(\mathbb {T}^1)}. \end{aligned}$$

Hence integration in time, (2.20) with $r=0$ and Young’s inequality finally yield (2.21).

In the case $r=1$ we use again that $\tilde{h}= \partial _s h$ solves (2.22). Testing this equation with $-\kappa \partial _s^2\tilde{h}$ yields in the same way as before

$$\begin{aligned}&\kappa \frac{d}{dt} \int _{\mathbb {T}^1} \frac{|\partial _s \tilde{h}|^2}{2}\, \textrm{d}s + \kappa ^2 \int _{\mathbb {T}^1} c_\kappa |\partial _s^2\tilde{h}|^2\,\textrm{d}s = \kappa \int _{\mathbb {T}^1} {\tilde{a}_\kappa } \partial _s \tilde{h} \partial _s^2 \tilde{h} \, \textrm{d}s -\int _{\mathbb {T}^1} (\partial _sg+\tilde{b}_\kappa \tilde{h}) \kappa \partial _s^2 h\,\textrm{d}s \end{aligned}$$

where

$$\begin{aligned}&\kappa \int _{\mathbb {T}^1} {\tilde{a}_\kappa }\partial _s \tilde{h} \partial _s^2 \tilde{h} \, \textrm{d}s = -\kappa \int _{\mathbb {T}^1} \partial _s {\tilde{a}_\kappa }\frac{|\partial _s \tilde{h}|^2}{2}\,\textrm{d}s \le C\kappa \Vert \tilde{h}(t)\Vert _{H^1(\mathbb {T}^1)}^2,\\&\quad -\int _{\mathbb {T}^1} (\partial _s g+\tilde{b}_\kappa \tilde{h}) \kappa \partial _s^2 \tilde{h}\,\textrm{d}s \le C(\Vert \partial _s g(t)\Vert _{L^2(\mathbb {T}^1)}+\Vert \partial _s h(t)\Vert _{L^2(\mathbb {T}^1)})\kappa \Vert \partial _s^2 \tilde{h}(t)\Vert _{L^2(\mathbb {T}^1)}. \end{aligned}$$

Therefore integration in time, (2.20) with $r=1$ and Young’s inequality yield (2.21) in the case $r=1$. Finally, the case $r\in (0,1)$ follows again by interpolation.

For the construction of the approximate solutions we will essentially use solution to the following linearized system:

$$\begin{aligned} \partial _t \textbf{w}_\varepsilon ^\pm +\textbf{w}_\varepsilon ^\pm \cdot \nabla \textbf{v}_0^\pm + \textbf{v}_0^\pm \cdot \nabla \textbf{w}_\varepsilon ^\pm -\nu ^\pm \Delta \textbf{w}_\varepsilon ^\pm + \nabla q_\varepsilon ^\pm&= \textbf{f}^\pm{} & {} \text {in}\Omega _t^{\varepsilon ,\pm }, t\in (0,T), \end{aligned}$$

(2.23)

$$\begin{aligned} {\text {div}}\textbf{w}_\varepsilon ^\pm&= 0{} & {} \text {in }\Omega _t^{\varepsilon ,\pm }, t\in (0,T),\end{aligned}$$

(2.24)

$$\begin{aligned} \llbracket \textbf{w}_\varepsilon \rrbracket =0,\quad \llbracket \nu D\textbf{w}_\varepsilon -q_\varepsilon \textbf{I}\rrbracket \cdot \textbf{n}_\varepsilon - \sigma (\Delta _{\Gamma _\varepsilon }h_\varepsilon )\circ S_\varepsilon \textbf{n}_\varepsilon&= 0{} & {} \text {on }\Gamma _t^\varepsilon , t\in (0,T),\end{aligned}$$

(2.25)

$$\begin{aligned} \textbf{w}_\varepsilon ^-|_{\partial \Omega }&= 0{} & {} \text {on }\partial \Omega \times (0,T),\end{aligned}$$

(2.26)

$$\begin{aligned} \textbf{w}_\varepsilon ^\pm |_{t=0}&=\textbf{w}_0{} & {} \text {in } \Omega ^{\varepsilon ,\pm }_0, \end{aligned}$$

(2.27)

together with

$$\begin{aligned}&\partial _t h_\varepsilon +(\textbf{n}_\varepsilon \cdot \textbf{w}_\varepsilon )\circ X_\varepsilon |_{r=0}+a_\varepsilon \partial _sh_\varepsilon + b_\varepsilon h_\varepsilon -\kappa _\varepsilon \Delta _{\Gamma _\varepsilon }h_\varepsilon \nonumber \\&\quad = (\textbf{n}_\varepsilon \cdot \textbf{u})\circ X_\varepsilon |_{r=0}\quad \text {in }\mathbb {T}^1\times (0,T), \end{aligned}$$

(2.28)

$$\begin{aligned}&h_\varepsilon |_{t=0} = h_0\quad \text {in } \mathbb {T}^1, \end{aligned}$$

(2.29)

where $\textbf{w}_\varepsilon ^\pm = \textbf{w}_\varepsilon |_{\Omega ^{\varepsilon ,\pm }}$, $q_\varepsilon ^\pm = q|_{\Omega ^{\varepsilon ,\pm }}$, $\textbf{n}_\varepsilon = \nabla d_\varepsilon$ and $\textbf{u}:\Omega \times (0,T)\rightarrow \mathbb {R}^2$ is given. Moreover, $T\in (0,T_0]$

$$\begin{aligned} \Omega ^{\varepsilon ,\pm }_t&:= \big (\Omega _t^\pm \setminus \Gamma _t(3\delta )\big )\cup \{x\in \Gamma _t(3\delta ): d_\varepsilon (x,t)\gtrless 0\},\quad \Omega ^{\varepsilon ,\pm }:= \bigcup _{t\in [0,T]}\Omega ^{\varepsilon ,\pm }_t\times \{t\},\\ \Gamma ^\varepsilon _t&:= \{x\in \Gamma _t (3\delta ): d_\varepsilon (x,t)=0\},\quad \Gamma ^\varepsilon := \bigcup _{t\in [0,T]}\Gamma ^\varepsilon _t\times \{t\} \end{aligned}$$

and

$$\begin{aligned} \Delta _{\Gamma _\varepsilon } h_\varepsilon (s,t):= (\Delta S_\varepsilon )(x,t)|_{x=X_\varepsilon (0,s,t)}\cdot \partial _s h_\varepsilon (s,t)+|\nabla S_\varepsilon (x,t)|^2|_{x=X_\varepsilon (0,s,t)} \partial _{s}^2 h_\varepsilon (s,t) \end{aligned}$$

for all $s\in \mathbb {T}^1$, $t\in (0,T)$. We note that by chain rule

$$\begin{aligned} \Delta (h_\varepsilon (S_\varepsilon (x,t),t))= \Delta _{\Gamma _\varepsilon } h_\varepsilon (S_\varepsilon (x,t),t) \qquad \text {for all }x\in \Gamma (3\delta ), t\in (0,T) \end{aligned}$$

for every sufficiently smooth $h_\varepsilon :\mathbb {T}^1\times (0,T)\rightarrow \mathbb {R}$.

More precisely, (2.23)–(2.25) are understood in the following weak sense:

$$\begin{aligned}&\langle \partial _t \textbf{w}_\varepsilon (t),\varvec{\varphi }\rangle _{V(\Omega )',V(\Omega )} + \int _{\Omega } (\textbf{w}_\varepsilon \cdot \nabla \textbf{v}_0+ \textbf{v}_0\cdot \nabla \textbf{w}_\varepsilon ) \cdot \varvec{\varphi }\, \textrm{d}x + \sum _{\pm } \int _{\Omega ^{\varepsilon ,\pm }_t(t)} \nu ^\pm D\textbf{w}_\varepsilon : D\varvec{\varphi }\, \textrm{d}x \nonumber \\&\quad = \langle \textbf{f}(t),\varvec{\varphi }\rangle _{V(\Omega )',V(\Omega )} + \sigma \int _{\Gamma ^\varepsilon _t} (\Delta _{\Gamma _\varepsilon }h_\varepsilon )\circ S_\varepsilon \textbf{n}_\varepsilon \cdot \varvec{\varphi }\,\textrm{d}\sigma \end{aligned}$$

(2.30)

for all $\varvec{\varphi }\in V(\Omega ):= H^1_0(\Omega )^2\cap L^2_\sigma (\Omega )$ and almost every $t\in (0,T)$, where as usual

$$\begin{aligned} L^2_\sigma (\Omega )= \overline{\{\varvec{\varphi }\in C_0^\infty (\Omega )^d: {\text {div}}\varvec{\varphi }=0\}}^{L^2(\Omega )^d}. \end{aligned}$$

Theorem 2.10

For $\varepsilon \in (0,1]$ let $\kappa _\varepsilon \in (0,1]$, $a_\varepsilon ,b_\varepsilon :\mathbb {T}^1\times [0,T]\rightarrow \mathbb {R}$ be continuously differentiable with uniformly bounded $C^1$-norms with respect to $\varepsilon \in (0,1]$. Then for every $\textbf{f}\in L^2(0,T;V(\Omega )')^d$, $\textbf{u}\in H^1(0,T;V(\Omega )')\cap L^2(0,T;V(\Omega ))$, $\textbf{w}_0\in L^2_\sigma (\Omega )$, and $h_0\in H^1(\mathbb {T}^1)$ there is a unique solution $\textbf{w}_\varepsilon \in H^1(0,T;V(\Omega )')\cap L^2(0,T;V(\Omega ))$, $h_\varepsilon \in H^1(0,T;L^2(\mathbb {T}^1))\cap L^2(0,T;H^2(\mathbb {T}^1))$ of (2.23)–(2.30). Moreover, there is some $C>0$ independent of $\varepsilon \in (0,1]$, h, g, $h_0$, and $T\in (0,T_0]$ such that

$$\begin{aligned}&\Vert h_\varepsilon \Vert _{L^\infty (0,T;H^1)} +\sqrt{\kappa _\varepsilon } \Vert \partial _s^2 h_\varepsilon \Vert _{L^2((0,T)\times \mathbb {T}^1)}+\Vert \textbf{w}_\varepsilon \Vert _{H^1(0,T;V(\Omega )')}+ \Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T;H^1)}\nonumber \\&\quad \ \le C\left( \Vert \textbf{f}\Vert _{L^2(0,T; V(\Omega )')}+\Vert \textbf{u}\Vert _{L^2(0,T;H^1)}+\Vert \partial _t \textbf{u}\Vert _{L^2(0,T;V(\Omega )')}+\Vert \textbf{w}_0\Vert _{L^2}+\Vert h_0\Vert _{H^1} \right) . \end{aligned}$$

(2.31)

Finally, if additionally $\textbf{f}\in L^2(0,T;L^2(\Omega )^2)$, $\textbf{w}_0\in V(\Omega )$, and $h_0\in H^{\frac{3}{2}}(\mathbb {T}^1)$, then

$$\begin{aligned}&\Vert h_\varepsilon \Vert _{H^1(0,T;H^{\frac{1}{2}})\cap L^2(0,T;H^{\frac{5}{2}})} +\Vert \textbf{w}_\varepsilon \Vert _{H^1(0,T;L^2)}+ \Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T;H^2(\Omega ^{\varepsilon ,\pm }_t))}\nonumber \\&\ \le \frac{C}{\kappa _\varepsilon }\left( \Vert \textbf{f}\Vert _{L^2((0,T)\times \Omega )}+\Vert \textbf{u}\Vert _{L^2(0,T;H^1)}+\Vert \partial _t \textbf{u}\Vert _{L^2(0,T;V(\Omega )')}+\Vert \textbf{w}_0\Vert _{H^1}+\Vert h_0\Vert _{H^1} \right) \end{aligned}$$

(2.32)

Proof

First of all existence of a unique solution $\textbf{w}_\varepsilon \in H^1(0,T;V(\Omega )')\cap L^2(0,T;V(\Omega ))$, $h_\varepsilon \in H^1(0,T; H^{-1}(\mathbb {T}^1))\cap L^2(0,T; H^1(\mathbb {T}^1))$ follows from the standard theory of abstract parabolic evolution equations for the Gelfand triple

$$\begin{aligned} V= V(\Omega )\times H^1(\mathbb {T}^1),\quad H= L^2_\sigma (\Omega )\times L^2(\mathbb {T}^1),\quad V'= V(\Omega )'\times H^{-1}(\mathbb {T}^1). \end{aligned}$$

Moreover, $h_\varepsilon \in H^1(0,T;L^2(\mathbb {T}^1))\cap L^2(0,T;H^2(\mathbb {T}^1))$ follows from standard regularity theory since $\textbf{n}_\varepsilon \cdot \textbf{w}_\varepsilon \circ X_\varepsilon |_{r=0}\in L^2(0,T; L^2(\mathbb {T}^1))$. Hence it only remains to show the uniform estimates.

Proof of (2.31): First of all we can reduce to the case $\textbf{u}=0$ simply by replacing $\textbf{w}$ by $\textbf{w}-\textbf{u}$ in the equations, where $\textbf{w}_0$ is replaced by $\textbf{w}_0-\textbf{u}|_{t=0}$ and $\textbf{f}$ has to be replaced by $\varvec{\tilde{f}}$ defined by

$$\begin{aligned}&\langle \varvec{\tilde{f}}(t),\varvec{\varphi }\rangle := \langle \textbf{f}(t),\varvec{\varphi }\rangle \\&\quad - \langle \partial _t \textbf{u}(t),\varvec{\varphi }\rangle _{V(\Omega )',V(\Omega )} + \int _{\Omega } (\textbf{u}\cdot \nabla \textbf{v}_0+ \textbf{v}_0\cdot \nabla \textbf{u}) \cdot \varvec{\varphi }\, \textrm{d}x + \sum _{\pm } \int _{\Omega ^{\varepsilon ,\pm }_t(t)} \nu ^\pm D\textbf{u} : D\varvec{\varphi }\, \textrm{d}x \end{aligned}$$

for all $\varvec{\varphi }\in V(\Omega )$ and $t\in (0,T)$. Adding $\textbf{u}$ afterwards to $\textbf{w}_\varepsilon$ yields the desired solution. Since

$$\begin{aligned} \Vert \varvec{\tilde{f}}\Vert _{L^2(0,T; V(\Omega )')}\le \Vert \textbf{f}\Vert _{L^2(0,T; V(\Omega )')}+ C\left( \Vert \textbf{u}\Vert _{L^2(0,T;H^1)}+\Vert \partial _t \textbf{u}\Vert _{L^2(0,T;V(\Omega )')}\right) \end{aligned}$$

one also obtains (2.31).

Now let $\textbf{u}\equiv 0$. Choosing $\varvec{\varphi }=\textbf{w}_\varepsilon$ in (2.30) and testing (2.28) with $|\partial _s X_\varepsilon (0,s,t)|\Delta _{\Gamma _\varepsilon } h_\varepsilon$ we obtain

$$\begin{aligned}&\frac{d}{dt}\left( \int _\Omega \frac{|\textbf{w}_\varepsilon (t)|^2}{2} \,\textrm{d}x + \int _{\mathbb {T}^1} \frac{|\nabla S_\varepsilon |^2|\partial _s h_\varepsilon (t)|^2}{2} |\partial _s X_\varepsilon (0,s,t)|\,\textrm{d}s \right) + \sum _{\pm }\int _{\Omega ^\pm } 2\nu ^\pm |D\textbf{w}_\varepsilon |^2\, \textrm{d}x\\&\qquad + \kappa _\varepsilon \int _{\mathbb {T}^1} |\partial _s X_\varepsilon (0,s,t)||\nabla S_\varepsilon |^4|\partial _s^2 h_\varepsilon (t)|^2\, \textrm{d}s= \langle \textbf{f},\varvec{\varphi }\rangle _{V(\Omega )',V(\Omega )} -\sum _{\pm }\int _{\Omega ^\pm (t)}\textbf{w}^\pm _\varepsilon \cdot \nabla \textbf{v}_0^\pm \cdot \textbf{w}_\varepsilon ^\pm \,\textrm{d}x\\&\qquad + \int _{\mathbb {T}^1}(\tilde{a}_\varepsilon \partial _s h_\varepsilon +\tilde{b}_\varepsilon h_\varepsilon )\partial _s^2 h_\varepsilon \, \textrm{d}s +\int _{\mathbb {T}^1} \left( \tilde{c}_\varepsilon \partial _s h_\varepsilon (t)+ \tilde{d}_\varepsilon h_\varepsilon (t)\right) \partial _s h_\varepsilon (t)\, \textrm{d}x\\&\quad \le \left( \Vert \textbf{f}\Vert _{L^2(\Omega )}+ \Vert \textbf{w}_\varepsilon \Vert _{L^2(\Omega )} + \Vert \textbf{a}\Vert _{H^{-\frac{1}{2}}(\Gamma _t^\varepsilon )} + \Vert (h_\varepsilon ,\partial _s h_\varepsilon )\Vert _{L^2(\mathbb {T}^1)} \right) \left( \Vert \textbf{w}_\varepsilon \Vert _{H^1(\Omega )}+ \Vert \partial _s h_\varepsilon \Vert _{L^2(\mathbb {T}^1)}\right) \end{aligned}$$

for some smooth and uniformly bounded $\tilde{a}_\varepsilon , \tilde{b}_\varepsilon , \tilde{c}_\varepsilon , \tilde{d}_\varepsilon :\mathbb {T}^1\times [0,T]\rightarrow \mathbb {R}$, where we note that

$$\begin{aligned} \int _{\Gamma _t^\varepsilon } \Delta _{\Gamma _\varepsilon } h_\varepsilon \circ S_\varepsilon \textbf{n}_\varepsilon \cdot \textbf{w}_\varepsilon \,\textrm{d}\sigma&= \int _{\mathbb {T}^1}(\textbf{n}_\varepsilon \cdot \textbf{w}_\varepsilon )(X_\varepsilon (0,s,t)) \Delta _{\Gamma _\varepsilon } h_\varepsilon (s,t) |\partial _s X_\varepsilon (0,s,t)| \,\textrm{d}s \quad \text {and}\\ \int _{\mathbb {T}^1}(\tilde{a}_\varepsilon \partial _s h_\varepsilon +\tilde{b}_\varepsilon h_\varepsilon )\partial _s^2 h_\varepsilon \, \textrm{d}s&= -\int _{\mathbb {T}^1}\left( (\partial _s\tilde{a}_\varepsilon ) \frac{|\partial _s h_\varepsilon |^2}{2}+\partial _s(\tilde{b}_\varepsilon h_\varepsilon )\partial _s h_\varepsilon \right) \, \textrm{d}s. \end{aligned}$$

Hence Young’s and Gronwall’s inequality yield the desired estimate (2.31).

Proof of (2.32): Now assume additionally that $\textbf{f}\in L^2(0,T;L^2(\Omega )^2)$, $\textbf{w}_0\in V(\Omega )$ and $h_0\in H^{\frac{3}{2}}(\mathbb {T}^1)$. (Note that we do not reduce to the case $\textbf{u}\equiv 0$ in this case since this is not compatible with the assumed regularity for $\textbf{u}$.) The estimate of $\Vert h_\varepsilon \Vert _{H^1(0,T;H^{\frac{1}{2}}(\mathbb {T}^1))\cap L^2(0,T;H^{\frac{5}{2}}(\mathbb {T}^1))}$ follows directly from (2.21) for $r=\frac{1}{2}$ and (2.31) using the equation (2.28). Hence

$$\begin{aligned}&\Vert \Delta _{\Gamma _\varepsilon }h\circ S_\varepsilon \Vert _{H^{\frac{1}{4}}(0,T;L^2(\Gamma _t^{\varepsilon }))\cap L^2(0,T;H^\frac{1}{2}(\Gamma _t^{\varepsilon })) }\\&\quad \le \frac{C}{\kappa _\varepsilon }\left( \Vert \textbf{f}\Vert _{L^2((0,T)\times \Omega )}+\Vert g\Vert _{H^{\frac{1}{4}}(0,T;L^2)\cap L^2(0,T;H^\frac{1}{2})}+\Vert \textbf{w}_0\Vert _{H^1}+\Vert h_0\Vert _{H^1} \right) \end{aligned}$$

Now the estimate of $\Vert \textbf{w}_\varepsilon \Vert _{H^1(0,T;L^2)}+ \Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T;H^2(\Omega ^{\varepsilon ,\pm }_t))}$ follows from standard estimates for the two-phase Stokes system, cf. e.g. [30] for the case that the interface $\Gamma ^\varepsilon _t$ is independent of $t\in (0,T)$. The result in the present case that $\Gamma _t^\varepsilon$ evolves smoothly with respect to t can be shown by the same perturbation argument as in the proof of Theorem A.14 in the appendix. $\square$

2.4 Spectral estimate

For the spectral estimate in $\varepsilon$-coordinates as in Sect. 2.1 let $\varepsilon _1>0$ be as in Theorem 2.1 and assume that (2.6) hold true. Moreover, we consider the rescaled variable

$$\begin{aligned} \rho _\varepsilon (x,t):=\frac{d_\varepsilon (x,t)}{\varepsilon }\quad \text { for }(x,t)\in \overline{\Gamma (3\delta )}. \end{aligned}$$

(2.33)

Finally, we assume the following structure of the approximate solution: let

$$\begin{aligned} \tilde{c}^A_\varepsilon (x,t):=\theta _0(\rho _\varepsilon (x,t))+ \varepsilon ^\mu p_\varepsilon (S_\varepsilon ,t)\theta _1(\rho _\varepsilon )+O(\varepsilon ^2), \end{aligned}$$

(2.34)

where $\mu \in [1,2)$ and $p_\varepsilon :\mathbb {T}^1\times [0,T_0]\rightarrow \mathbb {R}$ is measurable with $\Vert p_\varepsilon \Vert _\infty \le C$ and $\theta _1\in L^\infty (\mathbb {R})$ with

$$\begin{aligned} \int _\mathbb {R}f'''(\theta _0)(\theta _0')^2\theta _1\,\textrm{d}\rho =0. \end{aligned}$$

(2.35)

We set

$$\begin{aligned} c^A_\varepsilon :=\zeta (\frac{d_0}{\delta }) \tilde{c}^A_\varepsilon + (1-\zeta (\frac{d_0}{\delta }))(\chi _{\Omega ^+(t)}-\chi _{\Omega ^-(t)})\quad \text { in }\Omega , \end{aligned}$$

(2.36)

where $\zeta :\mathbb {R}\rightarrow \mathbb {R}$ is a smooth cutoff-function with $\zeta =1$ for $|r|\le 2$ and $\zeta =0$ for $|r|\ge \frac{5}{2}$.

The following spectral estimate will be a key ingredient for the proof of convergence.

Lemma 2.11

(Spectral estimate) Let the above assumptions in this section hold. Then there are some uniform $C_L,c_L>0$, $\varepsilon _0\in (0,\varepsilon _1]$ such that for every $\psi \in H^1(\Omega )$, $t\in [0,T_0]$, and $\varepsilon \in (0,\varepsilon _0]$ we have

$$\begin{aligned}{} & {} \int _{\Omega }\left( |\nabla \psi |^2+ \frac{f''(c^A_\varepsilon (.,t))}{\varepsilon ^2}\psi ^2\right) \,\textrm{d}x\\ {}{} & {} \quad \ge -C_L\Vert \psi \Vert _{L^2(\Omega )}^2 + \Vert \nabla \psi \Vert _{L^2(\Omega \backslash \Gamma ^\varepsilon _t(\frac{3\delta }{2}))}^2 + c_L\Vert \nabla _{{{\varvec{\tau }}_\varepsilon }}\psi \Vert _{L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))}^2, \end{aligned}$$

where $\nabla _{{\varvec{\tau }}_\varepsilon } \psi$ is as in (2.8) and $\Gamma ^\varepsilon _t(\frac{3\delta }{2}):={\Gamma ^\varepsilon (\frac{3\delta }{2})}\cap \left( \mathbb {R}^2\times \{t\}\right)$ with $X_\varepsilon$, $\Gamma ^\varepsilon (\frac{3\delta }{2}))$, and $\varepsilon _1$ are as in Theorem 2.1.

Remark 2.12

Because of (2.4) and $|\nabla \psi |\ge c|\nabla _{\tau _\varepsilon }\psi |$ (see e.g. the proof below), the result also holds for $\Gamma _t(2\delta )$ instead of $\Gamma ^\varepsilon _t(\frac{3\delta }{2})$ with a possibly smaller constant $c_L$.

Proof of Lemma 2.11

First, due to (2.4) and the definition (2.36) of $c^A_\varepsilon$, we obtain that for $\varepsilon >0$ small it holds

$$\begin{aligned} f''(c^A_\varepsilon )\ge 0\quad \text { in }\Gamma _\varepsilon (\tfrac{3\delta }{2}). \end{aligned}$$

Therefore let us first consider the integral over $\Gamma _\varepsilon (\frac{3\delta }{2})$: we can transform it into $(d_\varepsilon ,S_\varepsilon )$-coordinates and get

$$\begin{aligned} \int _{\Gamma _\varepsilon (\frac{3\delta }{2})}\left( |\nabla \psi |^2+ \frac{f''(c^A_\varepsilon )}{\varepsilon ^2}\psi ^2\right) \textrm{d}x=\int _{\mathbb {T}^1}\int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\left[ |\nabla \psi |^2\circ X_\varepsilon + \frac{f''(\tilde{c}^A_\varepsilon )}{\varepsilon ^2}\psi _\varepsilon ^2\right] J_\varepsilon \,\textrm{d}r\,\textrm{d}s, \end{aligned}$$

where we have set $\psi _\varepsilon :=\psi \circ X_\varepsilon$ and $J_\varepsilon$ is defined in (2.5). Via the chain rule we have the following transformation identity:

$$\begin{aligned} |\nabla \psi |^2\circ X_\varepsilon = (\nabla _{(r,s)}\psi _\varepsilon )^\top \left. \begin{pmatrix} |\nabla d_\varepsilon |^2 &{} \nabla d_\varepsilon \cdot \nabla S_\varepsilon \\ \nabla d_\varepsilon \cdot \nabla S_\varepsilon &{} |\nabla S_\varepsilon |^2 \end{pmatrix}\right| _{X_\varepsilon } \nabla _{(r,s)}\psi _\varepsilon . \end{aligned}$$

(2.37)

Therefore the asymptotics (2.6) together with Young’s inequality yields for $\varepsilon$ small

$$\begin{aligned} |\nabla \psi |^2\circ X_\varepsilon \ge (1-C\varepsilon ^2)|\partial _r\psi _\varepsilon |^2+\frac{1}{2}|\nabla _{\tau _\varepsilon }\psi |^2\circ X_\varepsilon . \end{aligned}$$

Altogether we obtain

$$\begin{aligned} \int _{\Omega }\left( |\nabla \psi |^2+ \frac{f''(c^A_\varepsilon (.,t))}{\varepsilon ^2}\psi ^2\right) \textrm{d}x&\ge \Vert \nabla \psi \Vert _{L^2(\Omega \backslash \Gamma _t^\varepsilon (\frac{3\delta }{2}))}^2+ \frac{1}{2} \Vert \nabla _{\tau _\varepsilon }\psi \Vert _{L^2(\Gamma _t^\varepsilon (\frac{3\delta }{2}))}^2 - C\Vert \psi \Vert ^2_{L^2(\Gamma _t^\varepsilon (\frac{3\delta }{2}))} \nonumber \\&\quad + (1-C\varepsilon ^2) \int _{\mathbb {T}^1}\int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\left[ |\partial _r\psi _\varepsilon |^2 + \frac{f''(\tilde{c}^A_\varepsilon (.,t))}{\varepsilon ^2}\psi _\varepsilon ^2\right] J_\varepsilon \,\textrm{d}r\,\textrm{d}s. \end{aligned}$$

(2.38)

The last term on the right hand side of (2.38) can be treated by well-known scaling and perturbation arguments as well as the spectral properties of differential operators on the real line similar to Chen [12]. More precisely, except for the dependency of $J_\varepsilon$ on $\varepsilon$ (which is not a problem because it is uniform in $\varepsilon$) the abstract 1D-spectral estimates in Moser [29, Section 5.1.3] are applicable after rescaling and yield the desired estimate. This shows the spectral estimate in Lemma 2.11. $\square$

Furthermore, we need more refined estimates of spectral decomposition type:

Corollary 2.13

Let the previous assumptions be valid and let $t\in \left[ 0,T_0\right]$ and $\psi \in H^1(\Gamma _t^\varepsilon (\frac{3\delta }{2}))$, where $\Gamma _t^\varepsilon (\frac{3\delta }{2})=\{x\in \Omega :(x,t)\in \Gamma ^\varepsilon (\frac{3\delta }{2})\}$ with $\Gamma ^\varepsilon (\frac{3\delta }{2})$ from Theorem 2.1. Moreover, let $\Lambda _{\varepsilon }\in \mathbb {R}$ be such that

$$\begin{aligned} \int _{\Gamma _t^\varepsilon (\frac{3\delta }{2})} \left| \nabla \psi \right| ^{2}+\frac{f''\left( c^A_\varepsilon (.,t)\right) }{\varepsilon ^2}\psi ^{2}\textrm{d}x\le \frac{\Lambda _{\varepsilon }}{\varepsilon } \end{aligned}$$

(2.39)

and denote $I_{\varepsilon }:=\left( -\frac{3\delta }{2\varepsilon },\frac{3\delta }{2\varepsilon }\right)$. Then, for $\varepsilon >0$ small enough, there exist functions $Z_\varepsilon \in H^{1}(\mathbb {T}^1)$, $\psi ^{\textbf{R}}_\varepsilon \in H^{1}(\Gamma _t^\varepsilon (\frac{3\delta }{2}))$ and smooth $\Psi _\varepsilon :I_{\varepsilon }\times \mathbb {T}^1\rightarrow \mathbb {R}$ such that

$$\begin{aligned} \psi (x)=\varepsilon ^{-\frac{1}{2}}Z_\varepsilon (s)\left( \beta _\varepsilon \theta _{0}'(\rho )+\Psi _\varepsilon (\rho ,s)\right) +\psi ^{\textbf{R}}_\varepsilon (x) \quad \text { for all } x\in \Gamma ^\varepsilon _t(\tfrac{3\delta }{2}) \end{aligned}$$

(2.40)

where

$$\begin{aligned} s=S_\varepsilon (x,t),\quad \rho = \frac{d_\varepsilon (x,t)}{\varepsilon }\end{aligned}$$

(2.41)

for almost all $x\in \Gamma _t^\varepsilon (\frac{3\delta }{2})$ and $\beta _\varepsilon :=\Vert \theta _0'\Vert _{L^2(I_\varepsilon )}^{-1}$. Moreover,

$$\begin{aligned}{} & {} \Vert \psi ^{\textbf{R}}_\varepsilon \Vert _{L^{2}\left( \Gamma _t^\varepsilon (\frac{3\delta }{4})\right) }^{2}\le C\left( \varepsilon \Lambda _{\varepsilon }+\varepsilon ^{2}\left\| \psi \right\| _{L^{2}\left( \Gamma _t^\varepsilon (\frac{3\delta }{2})\right) }^{2}\right) , \end{aligned}$$

(2.42)

$$\begin{aligned}{} & {} \Vert Z_\varepsilon \Vert _{H^{1}(\mathbb {T}^1)}^{2}+\Vert \nabla _{{\varvec{\tau }}_\varepsilon }\psi \Vert _{L^{2}(\Gamma _t^\varepsilon (\frac{3\delta }{2}))}^{2}+\Vert \psi ^{\textbf{R}}_\varepsilon \Vert _{H^{1}\left( \Gamma _t^\varepsilon (\frac{3\delta }{2})\right) }^{2} \le C\left( \left\| \psi \right\| _{L^{2}\left( \Gamma _t^\varepsilon (\frac{3\delta }{2})\right) }^{2}+\frac{\Lambda _{\varepsilon }}{\varepsilon }\right) ,\nonumber \\ \end{aligned}$$

(2.43)

and with $J_\varepsilon$ from (2.5)

$$\begin{aligned} \sup _{s\in \mathbb {T}^1}\left( \int _{I_{\varepsilon }}\left( \Psi _\varepsilon (\rho ,s)^{2}+\partial _\rho \Psi _\varepsilon (\rho ,s)^{2}\right) J_\varepsilon \left( \varepsilon \rho ,s,t\right) \textrm{d}\rho \right) \le C\varepsilon ^{2}. \end{aligned}$$

(2.44)

Proof

One can proceed similar to Abels, Marquardt [6, Corollary 2.12]. Here one uses the transformation into the $\varepsilon$-coordinates from Theorem 2.1 and spectral properties of 1D-differential operators on the real line similar to Chen [12], cf. also Moser [29, Section 5.1.3]. This yields the result with $\Vert Z_\varepsilon J_\varepsilon (0,.,t)^{\frac{1}{2}}\Vert _{H^1(\mathbb {T}^1)}$ instead of $\Vert Z_\varepsilon \Vert _{H^1(\mathbb {T}^1)}$ on the left hand side of (2.43). However, the additional factor is not a problem, because one can control $J_\varepsilon (0,.,t)^{-\frac{1}{2}}$ in $C^1(\mathbb {T}^1)$ independent of $\varepsilon$ using the form (2.5) and the assumptions on $d_\varepsilon$ and $S_\varepsilon$. Hence with the chain rule we obtain $\Vert Z_\varepsilon \Vert _{H^1(\mathbb {T}^1)}\le C \Vert Z_\varepsilon J_\varepsilon (0,.,t)^{\frac{1}{2}}\Vert _{H^1(\mathbb {T}^1)}$ and the result follows. $\square$

Remark 2.14

For $u\in H^1(\Gamma ^\varepsilon _t(\tfrac{3\delta }{2}))$ let us introduce the $\varepsilon$-dependent norms

$$\begin{aligned} \Vert u\Vert _{V_\varepsilon } =&\inf \left\{ \Vert Z\Vert _{H^1(\mathbb {T}^1)}+ \Vert v\Vert _{H^1(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))}+\frac{1}{\varepsilon }\Vert v\Vert _{L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))}: Z\in H^1(\mathbb {T}^1), v\in H^1(\Gamma ^\varepsilon _t(\tfrac{3\delta }{2})),\right. \\&\quad \qquad \left. u(x)=\frac{1}{\sqrt{\varepsilon }} Z(s)\theta _0'(\rho ) + v(x)\text { for all }x\in \Gamma ^\varepsilon _t(\tfrac{3\delta }{2}) \right\} , \end{aligned}$$

with the abbreviations from (2.41). Corollary 2.13 yields

$$\begin{aligned} \Vert u\Vert _{V_\varepsilon }^2+\Vert \nabla _{\tau _\varepsilon } u \Vert _{L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))}^2 \le C\left( \int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})} |\nabla u|^2 +\frac{1}{\varepsilon ^2} f''(c^A_\varepsilon (.,t))u^2\, \textrm{d}x + \Vert u\Vert _{L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))}^2 \right) . \end{aligned}$$

Here note that $\beta _\varepsilon$ from Corollary 2.13 is bounded uniformly for $\varepsilon$ small and in order to obtain the estimate one has to take care of the $\Psi _\varepsilon$-term from Corollary 2.13. However, this can be done by using the estimates in Corollary 2.13 and a rescaling argument.

We note that for every $\varepsilon >0$ the norm $\Vert .\Vert _{V_\varepsilon }$ is equivalent to the standard norm in $H^1(\tfrac{3\delta }{2}))$ (with $\varepsilon$-dependent constants). For the estimates of some critical remainder terms the choice of this norm will be essential. To estimate such remainder terms the following lemma will be used.

Lemma 2.15

Fix $t\in [0,T_0]$. Let $u\in H^1(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))$ and $r_\varepsilon :\Gamma ^\varepsilon _t(\frac{3\delta }{2})\rightarrow \mathbb {R}$ be a finite sum of terms of the form

$$\begin{aligned} a(\rho ) w_\varepsilon (S_\varepsilon ), \end{aligned}$$

where $a\in \mathcal {R}_{0,\alpha }$, $w_\varepsilon \in L^2(\mathbb {T}^1)$ and such that

$$\begin{aligned} \int _\mathbb {R}r_\varepsilon (\rho ,s,t) \theta '_0(\rho ) \, \textrm{d}\rho =0\qquad \text {for all }s\in \mathbb {T}^1. \end{aligned}$$

(2.45)

Then there are constants $C>0$, $\varepsilon _0\in (0,\varepsilon _1]$ independent of $t\in [0,T_0]$ such that for $\varepsilon \in (0,\varepsilon _0]$

$$\begin{aligned} \left| \int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})} r_\varepsilon u\,\textrm{d}x\right| \le C\varepsilon ^{\frac{3}{2}}\Vert w_\varepsilon \Vert _{L^2(\mathbb {T}^1)}\Vert u\Vert _{V_\varepsilon }. \end{aligned}$$

Proof

This can be done in the analogous way as in [3, Lemma 2.11]. $\square$

Remark 2.16

If we define dual norm

$$\begin{aligned} \Vert f\Vert _{V_\varepsilon '}:= \sup _{\Vert \varphi \Vert _{V_\varepsilon }\le 1} |\langle f,\varphi \rangle |\qquad \text {for }f\in (H^1(\tfrac{3\delta }{2})))', \end{aligned}$$

the lemma states that

$$\begin{aligned} \left\| \int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})} g_\varepsilon \cdot \,\textrm{d}x\right\| _{V_\varepsilon '} \le C\varepsilon ^{\frac{3}{2}}\Vert w_\varepsilon \Vert _{L^2(\mathbb {T}^1)}. \end{aligned}$$

3 Formally matched asymptotics

In this section we will discuss the construction of the approximate solutions except some higher order terms, which will be added in the next section. In comparision with previous works the main difference is that we obtain an expansion in terms of integer powers of $\varepsilon ^{\frac{1}{2}}$, which means we consider expansions in terms of $\varepsilon ^k$ with $k\in \frac{1}{2}\mathbb {N}_0$.

First of all we note that

$$\begin{aligned} {\text {div}}(\nabla c_\varepsilon \otimes \nabla c_\varepsilon )=\frac{1}{2}\nabla \big (|\nabla c_\varepsilon |^2\big )+\Delta c_\varepsilon \nabla c_\varepsilon . \end{aligned}$$

Therefore we can rewrite (1.1)–(1.3) as

$$\begin{aligned}&\partial _t \textbf{v}_\varepsilon +\textbf{v}_\varepsilon \cdot \nabla \textbf{v}_\varepsilon -{\text {div}}(2\nu (c_\varepsilon )D\textbf{v}_\varepsilon ) +\nabla p_\varepsilon = -\varepsilon \Delta c_\varepsilon \nabla c_\varepsilon , \end{aligned}$$

(3.1)

$$\begin{aligned}&{\text {div}}\textbf{v}_\varepsilon = 0, \end{aligned}$$

(3.2)

$$\begin{aligned}&\partial _t c_\varepsilon +\textbf{v}_\varepsilon \cdot \nabla c_\varepsilon =\varepsilon ^{\frac{1}{2}}\Delta c_{\epsilon }-\varepsilon ^{-\frac{3}{2}}f'(c_\varepsilon ) \end{aligned}$$

(3.3)

in $\Omega \times (0,T_0)$ by replacing $p_\varepsilon$ by $p_\varepsilon +\frac{1}{2}|\nabla c_\varepsilon |^2$.

3.1 The outer expansion

We assume that in $\Omega ^{\pm }\backslash \Gamma$ the solutions of (3.1)–(3.3) have the expansions

$$\begin{aligned} c_{\varepsilon }(x,t)&\approx \sum _{k\in \frac{1}{2}\mathbb {N}_0}\varepsilon ^{k}c_{k}^{\pm }(x,t), \ \ \textbf{v}_{\varepsilon }(x,t) \approx \sum _{k\in \frac{1}{2}\mathbb {N}_0}\varepsilon ^{k}\textbf{v}_{k}^{\pm }(x,t),\ \\ p_{\varepsilon }(x,t)&\approx \sum _{k\in \frac{1}{2}\mathbb {N}_{-2}}\varepsilon ^{k}p_{k}^{\pm }(x,t), \end{aligned}$$

where $\mathbb {N}_{-2}=\mathbb {N}_0\cup \{-1,-2\}$ and $c_{k}^{\pm }$, $\textbf{v}_{k}^{\pm }$ and $p_{k}^{\pm }$ are smooth functions defined in $\Omega ^{\pm }$. Here $\varphi _\varepsilon (x,t)\approx \sum _{k\ge 0,k\in \frac{1}{2}\mathbb {N}_0} \varepsilon ^k \varphi _k^\pm (x,t)$ for $\varphi _\varepsilon =c_\varepsilon , \textbf{v}_\varepsilon$ (analogously for $p_\varepsilon$) is understood in the sense that for any $N\in \frac{1}{2}\mathbb {N}_0$ we have

$$\begin{aligned} \varphi _\varepsilon (x,t)- \sum _{ k\in \frac{1}{2}\mathbb {N}_0,k\le N} \varepsilon ^k\varphi ^\pm _k(x,t) = O(\varepsilon ^{N+\frac{1}{2}}) \qquad \text {in }\Omega ^\pm \end{aligned}$$

and the same if $\varphi _\varepsilon$ and $\varphi _k$ are replaced by $\partial _t^j\partial _x^\alpha \varphi _\varepsilon$ and $\partial _t^j\partial _x^\alpha \varphi _k^\pm$, respectively, for any $j\in \mathbb {N}_0$, $\alpha \in \mathbb {N}_0^n$.

Plugging this ansatz into (3.1)–(3.3) and (1.4), using a Taylor expansion for $f'$ and $\nu$, the Dirichlet boundary condition for $c_\varepsilon$, and matching the $\mathcal {O}(\varepsilon ^{-1}),\mathcal {O}(\varepsilon ^{-\frac{1}{2}})$ terms one obtains in a standard manner (cf. e.g. [3, Appendix])

$$\begin{aligned} c_{0}^{\pm }=\pm 1, \quad c_k^\pm = 0 \quad \text {for }k\ge \frac{1}{2}, \quad \nabla p_{-1}^{\pm }=\nabla p_{-\frac{1}{2}}^{\pm } =0\qquad \text {in }\Omega ^\pm \end{aligned}$$

(3.4)

and

$$\begin{aligned} \partial _t\textbf{v}_{k}^{\pm }+\textbf{v}_{0}^{\pm }\cdot \nabla \textbf{v}_{k}^{\pm }+\textbf{v}_{k}^{\pm }\cdot \nabla \textbf{v}_{0}^{\pm } -\nu ^{\pm }\Delta \textbf{v}_{k}^{\pm }+\nabla p_{k}^{\pm }&=-\sum _{j\in \frac{1}{2} \mathbb {N}, \frac{1}{2}\le j\le k-\frac{1}{2}}\textbf{v}_{j}^{\pm }\cdot \nabla \textbf{v}_{k-j}^{\pm }{} & {} \text {in }\Omega ^\pm , \end{aligned}$$

(3.5)

$$\begin{aligned} {\text {div}}\textbf{v}_{k}^{\pm }&=0{} & {} \text {in }\Omega ^\pm , \end{aligned}$$

(3.6)

$$\begin{aligned} {\textbf{v}_k^-|_{\partial \Omega }}&=0{} & {} \text {on }\partial \Omega \end{aligned}$$

(3.7)

for every $k\ge 0$. For simplicity we take

$$\begin{aligned} p_{-1}^{\pm }= p_{-\frac{1}{2}}^{\pm } =0. \end{aligned}$$

(3.8)

Remark 3.1

As in [3, 5] we extend $(c_{k}^{\pm },\textbf{v}_{k}^{\pm },p_{k}^{\pm })$, $k\ge 0$, defined on $\Omega ^{\pm }$, to $\Omega ^{\pm }\cup \Gamma (3\delta )$ such that ${\text {div}}\textbf{v}_k^{\pm }=0$ in $\Omega ^{\pm }\cup \Gamma (3\delta )$ for all $k\ge 0$. We refer to [3, Remark A.1] for the details.

For the following we define

$$\begin{aligned} \textbf{W}_{k}^{\pm }(x,t)&=\partial _t\textbf{v}_{k}^{\pm }(x,t)+\sum _{j\in \frac{1}{2} \mathbb {N}_0, j\le k}\textbf{v}_{k-j}^{\pm }\cdot \nabla \textbf{v}_{j}^{\pm }-\nu ^\pm \Delta \textbf{v}_{k}^{\pm }(x,t)+\nabla p_{k}^{\pm }(x,t) \end{aligned}$$

(3.9)

$$\begin{aligned} \textbf{W}^{\pm }&=\sum _{k\in \frac{1}{2}\mathbb {N}_0}\varepsilon ^{k}\textbf{W}_{k}^{\pm }, \end{aligned}$$

(3.10)

for $(x,t)\in \Omega ^{\pm }\cup \Gamma (2\delta )$. Because of (3.5), it holds $\textbf{W}_{k}^{\pm }(x,t)=0$ for all $(x,t)\in \overline{\Omega ^{\pm }}.$

3.2 The inner expansion

Close to the interface $\Gamma$ we introduce a stretched variable

$$\begin{aligned} \rho =\rho (x,t):=\frac{d_\varepsilon (x,t)}{\varepsilon }\qquad \text {for all }(x,t)\in \Gamma (3\delta ) \end{aligned}$$

(3.11)

for $\varepsilon \in \left( 0,1\right)$, where $d_{\varepsilon }:\Gamma (3\delta )\rightarrow \mathbb {R}$ satisfies

$$\begin{aligned} |\nabla d_\varepsilon (x,t)|^2 \approx 1\qquad \text {for all }(x,t)\in \Gamma (3\delta ), \end{aligned}$$

(3.12)

which has to be understood as $|\nabla d_\varepsilon (x,t)|^2=1+ O(\varepsilon ^{N+\frac{1}{2}})$ for any $N\in \frac{1}{2}\mathbb {N}_0$ similary as before. Formally, $d_\varepsilon$ is the signed distance function to $\Gamma ^\varepsilon$, which is the 0-level set of $c_{\varepsilon }$. Moreover, we assume the asymptotic expansion

$$\begin{aligned} d_\varepsilon (x,t) \approx \sum _{k\in \frac{1}{2}\mathbb {N}_0} \varepsilon ^kd_k(x,t) \qquad \text {for }(x,t)\in \Gamma (3\delta ) \end{aligned}$$

understood in the same way as before, where $d_0(x,t)=d_{\Gamma _t}(x)$ for all $(x,t)\in \Gamma (3\delta )$. Here and in the following we assume already that $(\textbf{v}_0^\pm ,p_0^\pm ,\Gamma )$ is a smooth solution of (1.6)–(1.12), although these equations can also be derived throughout the formal expansion. Since for the asymptotic expansion as $\varepsilon \rightarrow 0$ only small values of $\varepsilon >0$ matter, we may assume that

$$\begin{aligned} |d_\varepsilon (x,t)-d_0(x,t)-\sqrt{\varepsilon } d_{1/2}(x,t)|\le M_0\varepsilon \qquad \text {for all }(x,t)\in \Gamma (3\delta ) \end{aligned}$$

(3.13)

for some $M_0>0$. Moreover, we choose $\eta :\mathbb {R}\rightarrow [0,1]$ such that $\eta =0$ in $(-\infty ,-1]$, $\eta =1$ in $[1,\infty )$, $\eta -\frac{1}{2}$ is odd and $\eta '\ge 0$ in $\mathbb {R}$. Then we have by integration by parts

$$\begin{aligned} \int _{-\infty }^{+\infty }\nu (\theta _0)\eta '(\rho )\,\textrm{d}\rho = \left. \left[ \nu (\theta _0(\rho )) \left( \eta (\rho )-\frac{1}{2}\right) \right] \right| _{\rho =-\infty }^\infty = \frac{\nu ^++\nu ^-}{2}=: \overline{\nu } \end{aligned}$$

(3.14)

since $\nu '(\theta _0)$ is even by the assumptions on $\nu '$. Furthermore, we define

$$\begin{aligned} \eta ^{\varepsilon ,\pm }(\rho , x,t)&=\eta (-M-1\pm \rho \mp \varepsilon ^{-\frac{1}{2}}d_{1/2}(x,t))\quad \text {for }\rho \in \mathbb {R}, (x,t)\in \Gamma (3\delta ). \end{aligned}$$

Remark 3.2

In the following we will insert terms $\textbf{W}^{\pm }\eta ^{\varepsilon ,\pm }$ in the equation to ensure some matching conditions. We have to make sure that these terms vanish if $\rho = \frac{d_\varepsilon (x,t)}{\varepsilon }$. Because of (3.13), we have for $\rho =\frac{d_{\varepsilon }(x,t)}{\varepsilon }$ and $(x,t)\in \Gamma (3\delta )$ with $d_{\Gamma }(x,t)\ge 0$ that $\rho -\varepsilon ^{-\frac{1}{2}}d_{1/2}(x,t)\ge -M$. Hence $\eta ^{\varepsilon ,-}(\rho ,x,t)=0$ and, since $(x,t)\in \overline{\Omega ^{+}}$, we have $\textbf{W}^{+}(x,t)=0$. Altogether

$$\begin{aligned} \textbf{W}^{+}\eta ^{\varepsilon ,+}+\textbf{W}^{-}\eta ^{\varepsilon ,-}=0\quad \text {in }\overline{\Omega ^+}. \end{aligned}$$

In the same way one shows this in $\overline{\Omega ^-}$.

For the inner expansion we use the ansatz

$$\begin{aligned} c_{\varepsilon }(x,t) =\hat{c}_{\varepsilon }\big (\rho ,x,t\big ),\quad \textbf{v}_{\varepsilon }(x,t)=\hat{\textbf{v}}_{\varepsilon }\big (\rho ,x,t\big ),\quad p_{\varepsilon }(x,t) =\hat{p}_{\varepsilon }\big (\rho ,x,t\big ) \end{aligned}$$

for $(x,t)\in \Gamma (3\delta )$, where $\rho \in \mathbb {R}$ is as in (3.11), and use it in the expansion of (3.1)–(3.3) for smooth $\hat{c}_{\varepsilon },\hat{p}_{\varepsilon }:\mathbb {R}\times \Gamma (3\delta )\rightarrow \mathbb {R}$, $\hat{\textbf{v}}_{\varepsilon }:\mathbb {R}\times \Gamma (3\delta )\rightarrow \mathbb {R}^{2}$.

We use that

$$\begin{aligned}&{\text {div}}(2\nu (c_\varepsilon )D\textbf{v}_\varepsilon )= \varepsilon ^{-2}\partial _{\rho }\big (\nu (\hat{c}_\varepsilon )\partial _{\rho }\hat{\textbf{v}}_\varepsilon \big )-\varepsilon ^{-1}\partial _{\rho }\big (\nu (\hat{c}_\varepsilon ){\text {div}}\hat{\textbf{v}}_\varepsilon \big )\nabla d_{\varepsilon }\nonumber \\&\qquad +\varepsilon ^{-1}\partial _{\rho }\big (2\nu (\hat{c}_\varepsilon )D\hat{\textbf{v}}_\varepsilon \big )\big )\cdot \nabla d_{\varepsilon } +\varepsilon ^{-1}{\text {div}}\big (2\nu (\hat{c}_\varepsilon )D_d\hat{\textbf{v}}_\varepsilon \big )+{\text {div}}\big (2\nu (\hat{c}_\varepsilon )D\hat{\textbf{v}}_\varepsilon \big ), \end{aligned}$$

(3.15)

provided (3.2) holds, where

$$\begin{aligned} D\hat{\textbf{v}}_\varepsilon&=\frac{1}{2}(\nabla \hat{\textbf{v}}_\varepsilon +(\nabla \hat{\textbf{v}}_\varepsilon )^T),\quad D_d\hat{\textbf{v}}_\varepsilon =\frac{1}{2}\big (\partial _\rho \hat{\textbf{v}}_\varepsilon \otimes \nabla d_{\varepsilon }+(\partial _\rho \hat{\textbf{v}}_\varepsilon \otimes \nabla d_{\varepsilon })^T\big ). \end{aligned}$$

Moreover, we note that

$$\begin{aligned} \nabla c_{\varepsilon }&=\varepsilon ^{-1}\partial _{\rho }\hat{c}_{\varepsilon }\nabla d_{\varepsilon }+\nabla \hat{c}_{\varepsilon }, \\ \Delta c_{\varepsilon }&=\varepsilon ^{-2}\partial _{\rho \rho }\hat{c}_{\varepsilon }+2\varepsilon ^{-1}\nabla \partial _{\rho }\hat{c}_{\varepsilon }\cdot \nabla d_{\varepsilon }+\varepsilon ^{-1}\partial _{\rho }\hat{c}_{\varepsilon }\Delta d_{\varepsilon }+\Delta \hat{c}_{\varepsilon }, \end{aligned}$$

where we have used (3.12). This yields

$$\begin{aligned} \varepsilon \Delta c_{\varepsilon }\nabla c_{\varepsilon } =\varepsilon ^{-2}\partial _{\rho }\hat{c}_{\varepsilon }\partial _{\rho \rho }\hat{c}_{\varepsilon }\nabla d_{\varepsilon } +\varepsilon ^{-1}\mathbb {A}_{\varepsilon }+\mathbb {B}_{\varepsilon }+\varepsilon \mathbb {C}_{\varepsilon }, \end{aligned}$$

(3.16)

where

$$\begin{aligned} \mathbb {A}_{\varepsilon }&=\partial _{\rho \rho }\hat{c}_{\varepsilon }\nabla \hat{c}_{\varepsilon }+2\nabla \partial _{\rho }\hat{c}_{\varepsilon }\cdot \nabla d_{\varepsilon }\partial _{\rho }\hat{c}_{\varepsilon }\nabla d_{\varepsilon } +\big (\partial _{\rho }\hat{c}_{\varepsilon }\big )^2\Delta d_{\varepsilon }\nabla d_{\varepsilon },\nonumber \\ \mathbb {B}_{\varepsilon }&=2\nabla \partial _{\rho }\hat{c}_{\varepsilon }\cdot \nabla d_{\varepsilon }\nabla \hat{c}_{\varepsilon }+\partial _{\rho }\hat{c}_{\varepsilon }\Delta d_{\varepsilon }\nabla \hat{c}_{\varepsilon }+\partial _{\rho }\hat{c}_{\varepsilon }\Delta \hat{c}_{\varepsilon }\nabla d_{\varepsilon }, \nonumber \\ \mathbb {C}_{\varepsilon }&=\Delta \hat{c}_{\varepsilon }\nabla \hat{c}_{\varepsilon }.\nonumber \end{aligned}$$

(3.17)

Hence for $\rho$ as in (3.11) the system (3.1)–(3.3) is equivalent to

$$\begin{aligned} \partial _{\rho }\big (\nu (\hat{c}_{\varepsilon })\partial _{\rho }\hat{\textbf{v}}_\varepsilon \big )&= 2\partial _{\rho }\hat{c}_{\varepsilon }\partial _{\rho \rho }\hat{c}_{\varepsilon }\nabla d_{\varepsilon } + \varepsilon \bigg (\partial _{\rho }\hat{\textbf{v}}_\varepsilon \partial _{t}d_{\varepsilon }+\hat{\textbf{v}}_\varepsilon \cdot \nabla d_{\varepsilon }\partial _{\rho }\hat{\textbf{v}}_\varepsilon +\partial _{\rho }\hat{p}_\varepsilon \nabla d_{\varepsilon }+\mathbb {A}_{\varepsilon }\nonumber \\&\quad -\partial _{\rho }\big (2\nu (\hat{c}_\varepsilon )D\hat{\textbf{v}}_\varepsilon \big )\cdot \nabla d_{\varepsilon } -{\text {div}}\big (2\nu (\hat{c}_\varepsilon )D_d\hat{\textbf{v}}_\varepsilon \big )+\partial _{\rho }\big (\nu (\hat{c}_\varepsilon ){\text {div}}\hat{\textbf{v}}_\varepsilon \big )\nabla d_{\varepsilon }\bigg ) \nonumber \\&\quad +\varepsilon ^2\bigg (\partial _{t}\hat{\textbf{v}}_\varepsilon +\hat{\textbf{v}}_\varepsilon \cdot \nabla \hat{\textbf{v}}_\varepsilon +\nabla \hat{p}_\varepsilon -{\text {div}}\big (2\nu (\hat{c}_\varepsilon )D\hat{\textbf{v}}_\varepsilon \big )+\textbf{W}^{+}\eta ^{\varepsilon ,+}+\textbf{W}^{-}\eta ^{\varepsilon ,-}\bigg ) \nonumber \\ {}&\quad +\varepsilon ^2\mathbb {B}_{\varepsilon }+\varepsilon ^3\mathbb {C}_{\varepsilon } +\big (\nu (\theta _0)\eta '(\rho )\big )'\textbf{u}_{\varepsilon }\big (d_\varepsilon -\varepsilon \rho \big ) +\varepsilon \textbf{l}_{\varepsilon }\eta '(\rho )\big (d_\varepsilon -\varepsilon \rho \big ), \end{aligned}$$

(3.18)

$$\begin{aligned} \partial _{\rho }\hat{\textbf{v}}_\varepsilon \cdot \textbf{n}_\varepsilon&= -\varepsilon {\text {div}}_x\hat{\textbf{v}}_\varepsilon +\mathbf {u_{\varepsilon }}\cdot \nabla d_{\varepsilon }\eta '(\rho )\big (d_{\varepsilon }-\varepsilon \rho \big ), \end{aligned}$$

(3.19)

$$\begin{aligned} \partial ^2_{\rho }\hat{c}_\varepsilon -f'(\hat{c}_{\varepsilon })&= \varepsilon ^{\frac{1}{2}}\bigg (\partial _{\rho }\hat{c}_\varepsilon \partial _t d_{\varepsilon }+\partial _{\rho }\hat{c}_\varepsilon \hat{\textbf{v}}_\varepsilon \cdot \nabla d_{\varepsilon }\bigg ) -\varepsilon \bigg (\partial _{\rho }\hat{c}_\varepsilon \Delta d_{\varepsilon }+2\nabla \partial _{\rho }\hat{c}_\varepsilon \cdot \nabla d_{\varepsilon }\bigg ) \nonumber \\&\quad +\varepsilon ^{\frac{3}{2}}\bigg (\partial _t\hat{c}_\varepsilon +\hat{\textbf{v}}_\varepsilon \cdot \nabla \hat{c}_\varepsilon \bigg ) -\varepsilon ^2\Delta \hat{c}_\varepsilon +\varepsilon ^{\frac{1}{2}}\hat{g}_{\varepsilon }\left( d_{\varepsilon }-\varepsilon \rho \right) . \end{aligned}$$

(3.20)

We note that by the definition of $\textbf{W}^\pm$ the right-hand side of (3.17) converges exponentially to zero as $|\rho |\rightarrow \infty$. Here we introduced $\textbf{u}_{\varepsilon }(x,t)$ and $\textbf{l}_{\varepsilon }(x,t)$ for $(x,t)\in \Gamma (3\delta )$ in a similar manner as in [8]. $\textbf{l}_\varepsilon$ will ensure the compatibility conditions in $\Gamma (3\delta )\backslash \Gamma$ for (3.17) and $\textbf{u}_\varepsilon$ will ensure the matching conditions for $\textbf{v}_\varepsilon$ on $\Gamma (3\delta )\backslash \Gamma$. Moreover, an auxiliary function $\hat{g}_{\varepsilon }(\rho ,x,t)=\varepsilon g_{\varepsilon }\eta '(\rho )-\theta _0'(\rho )\hat{\phi }_\varepsilon$ is introduced, which is used to satisfy the compatibility conditions in $\Gamma (3\delta )\backslash \Gamma$ for (3.19). Note that the extra terms vanish on the relevant set

$$\begin{aligned} S^{\varepsilon }:=\big \{ (\rho ,x,t)\in \mathbb {R}\times \Gamma (3\delta ):\rho =\tfrac{d_{\varepsilon }(x,t)}{\varepsilon }\big \}. \end{aligned}$$

By the definition of $\hat{g}_\varepsilon$ we have

$$\begin{aligned} \varepsilon ^{\frac{1}{2}}\hat{g}_{\varepsilon }\left( d_{\varepsilon }-\varepsilon \rho \right) =-\varepsilon ^{\frac{5}{2}}\eta '(\rho )\rho g_{\varepsilon }+\varepsilon ^{\frac{3}{2}}\big (\eta '(\rho ) g_{\varepsilon }d_{\varepsilon }+\theta _0'(\rho )\rho \phi ^\varepsilon \big )-\varepsilon ^{\frac{1}{2}}\theta _0'(\rho )\hat{\phi }^\varepsilon d_{\varepsilon }. \end{aligned}$$

More precisely, we choose the auxiliary functions to have expansions of the form

$$\begin{aligned} \hat{\phi }_\varepsilon (\rho ,x,t)=\hat{\phi }_{0}(\rho ,x,t)+\varepsilon ^{\frac{1}{2}}\hat{\phi }_{\frac{1}{2}}(\rho ,x,t)\quad \text {in }\Gamma (3\delta ) \end{aligned}$$

with

$$\begin{aligned} \hat{\phi }_0&=\frac{\partial _t d_{\Gamma }+\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }}{ d_{\Gamma }}, \end{aligned}$$

(3.21)

$$\begin{aligned} \hat{\phi }_{\frac{1}{2}}&=\frac{\partial _t d_{\frac{1}{2}}+\hat{\textbf{v}}_0\cdot \nabla d_{\frac{1}{2}}+\hat{\textbf{v}}_{\frac{1}{2}}\cdot \nabla d_{\Gamma }-\hat{\phi }_0d_{\frac{1}{2}}-\Delta d_{\Gamma }}{ d_{\Gamma }}, \end{aligned}$$

(3.22)

and

$$\begin{aligned} \textbf{u}_{\varepsilon }(x,t) \approx \sum _{k\in \frac{1}{2}\mathbb {N}_0}\textbf{u}_{k}(x,t)\varepsilon ^{k},\quad g_{\varepsilon }(x,t) \approx \sum _{k\in \frac{1}{2}\mathbb {N}_0}g_{k}(x,t)\varepsilon ^{k}, \quad \textbf{l}_{\varepsilon }(x,t) \approx \sum _{k\in \frac{1}{2}\mathbb {N}_0}\textbf{l}_{k}(x,t)\varepsilon ^{k} \end{aligned}$$

(3.23)

for $(x,t)\in \Gamma (3\delta )$. We note that $\hat{\phi }_{0},\hat{\phi }_{\frac{1}{2}}$ are defined on $\Gamma$ as limits $d_\Gamma \rightarrow 0$, which exist due to (1.10), (3.30), and since it will turn out that $\hat{\textbf{v}}|_{\Gamma }=\textbf{v}_0^\pm |_{\Gamma }$, cf. (3.53) below. In particular,

$$\begin{aligned} \hat{\phi }_0&=\nabla \big (\partial _t d_{\Gamma }+\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }\big )\cdot \nabla d_{\Gamma }=\nabla (\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma })\cdot \nabla d_{\Gamma }&\quad&\text {on}\ \Gamma , \end{aligned}$$

(3.24)

$$\begin{aligned} \hat{\phi }_{\frac{1}{2}}&=\nabla \big (\partial _t d_{\frac{1}{2}}+\hat{\textbf{v}}_0\cdot \nabla d_{\frac{1}{2}}+\hat{\textbf{v}}_{\frac{1}{2}}\cdot \nabla d_{\Gamma }-\hat{\phi }_0d_{\frac{1}{2}}-\Delta d_{\Gamma }\big )\cdot \nabla d_{\Gamma }&\quad&\text {on}\ \Gamma \end{aligned}$$

(3.25)

since $\nabla d_\Gamma \cdot \nabla \partial _t d_\Gamma = \frac{1}{2} \partial _t |\nabla d_\Gamma |^2=0$. Here, in order to obtain (3.1)–(3.3) (approximately), the equations above only have to hold in $S^{\varepsilon }=\big \{ (\rho ,x,t)\in \mathbb {R}\times \Gamma (3\delta ):\rho =\tfrac{d_{\varepsilon }(x,t)}{\varepsilon }\big \}.$ But in the following we consider them as ordinary differential equations in $\rho \in \mathbb {R}$, where $(x,t)\in \Gamma (3\delta )$ are seen as fixed parameters. Thus we require from now on that (3.17)–(3.19) are fulfilled even for all $(\rho ,x,t)\in \mathbb {R}\times \Gamma (3\delta )$.

Furthermore, we assume that we have the expansions

$$\begin{aligned} \hat{c}_{\varepsilon }(\rho ,x,t)&\approx \sum _{k\in \frac{1}{2}\mathbb {N}_0}\varepsilon ^{k}\hat{c}_{k}(\rho ,x,t),\quad \hat{\textbf{v}}_{\varepsilon }(\rho ,x,t) \approx \sum _{k\in \frac{1}{2}\mathbb {N}_0}\varepsilon ^{k}\hat{\textbf{v}}_{k}(\rho ,x,t),\quad \\ \hat{p}_{\varepsilon }(\rho ,x,t)&\approx \sum _{k\in \frac{1}{2}\mathbb {N}_{-2}}\varepsilon ^{k}\hat{p}_{k}(\rho ,x,t). \end{aligned}$$

understood in the same way as before. Actually, in the expansion it turns out that $\hat{c}_0=\theta _0$ and $\hat{c}_{\frac{1}{2}}=\hat{c}_1=0$. To simplify the following presentation we already assume $\hat{c}_{\frac{1}{2}}=\hat{c}_1=0$. As usual we normalize $\hat{c}_{k}$ such that

$$\begin{aligned} \hat{c}_{k}(0,x,t)=0\qquad \text {for all }(x,t)\in \Gamma (3\delta ), k\ge 0. \end{aligned}$$

(3.26)

In order to match the inner and outer expansions, we require that for all k the so-called inner–outer matching conditions

$$\begin{aligned} \sup _{(x,t)\in \Gamma (3\delta )}\left| \partial _{x}^{m}\partial _{t}^{n}\partial _{\rho }^{l}\left( \varphi (\pm \rho ,x,t)-\varphi ^{\pm }(x,t)\right) \right|&\le Ce^{-\alpha \rho }, \end{aligned}$$

(3.27)

where $\varphi =\hat{c}_{k},\hat{\textbf{v}}_{k}$ with $k\ge 0$ and $\hat{p}_{k}$ with $k\ge -1$ hold for constants $\alpha ,C>0$ and all $\rho >0$, $m,n,l\ge 0$.

For the non-linear terms $\Phi =\nu ,f'$ we use the expansion

$$\begin{aligned} \Phi (c_\varepsilon )&=\Phi (\hat{c}_{0})+\sum _{k=\frac{3}{2}}^{N+2}\varepsilon ^k \Phi '(\hat{c}_{0})\hat{c}_{k} +\sum _{k=\frac{5}{2}}^{N+2}\varepsilon ^{k}\Phi _{k-1}(\hat{c}_{0},\hat{c}_{\frac{3}{2}},\ldots ,\hat{c}_{k-1}) \nonumber \\ {}&\quad +\varepsilon ^{N+\frac{5}{2}}\Phi _{N+\frac{3}{2}}(c_\varepsilon , \hat{c}_{0},\hat{c}_{\frac{3}{2}},\ldots ,\hat{c}_{N+\frac{3}{2}}) \end{aligned}$$

(3.28)

where we have used $\hat{c}_{1/2}=\hat{c}_1=0$. Here $\Phi _{k-1}(\hat{c}_{0},\hat{c}_{\frac{3}{2}},\ldots ,\hat{c}_{k-1})$ and $\Phi _{N+\frac{3}{2}}(c_\varepsilon , \hat{c}_{0},\hat{c}_{\frac{3}{2}},\ldots ,\hat{c}_{N+\frac{3}{2}})$ are polynomials in $\hat{c}_{\frac{3}{2}},\ldots , \hat{c}_{k-1}$ with coefficients that depend smoothly on $\hat{c}_0$ and $(\hat{c}_0,c_\varepsilon )$, respectively.

Using this expansion in (3.12) we obtain

$$\begin{aligned} |\nabla d_\varepsilon |^2\approx&\underbrace{|\nabla d_0|^2}_{=1} + 2\varepsilon ^{\frac{1}{2}} \nabla d_0\cdot \nabla d_{1/2} +\sum \limits _{ k\ge 1,k\in \frac{1}{2}\mathbb {N}_0}\varepsilon ^{k}\sum \limits _{0\le i\le k,i\in \frac{1}{2}\mathbb {N}_0}\nabla d_{i}\cdot \nabla d_{k-i}. \end{aligned}$$

Hence, in order to satisfy (3.12) (up to higher order terms in $\varepsilon$) in ${\Gamma (3\delta )}$ we choose $d_k$, $k=\frac{1}{2}, 1, \frac{3}{2},\ldots$ successively such that

$$\begin{aligned} \nabla d_0\cdot \nabla d_{1/2}&=0,{} & {} \text {in }\Gamma (3\delta ), \end{aligned}$$

(3.29)

$$\begin{aligned} \nabla d_0\cdot \nabla d_{k}&=-\frac{1}{2}\sum \limits _{\frac{1}{2}\le i\le k-\frac{1}{2},i\in \frac{1}{2}\mathbb {N}_0}\nabla d_{i}\cdot \nabla d_{k-i}\quad \text {for } k\ge 1,\ k\in \frac{1}{2}\mathbb {N}_0,{} & {} \text {in}\,\Gamma (3\delta ). \end{aligned}$$

(3.30)

Furthermore, we choose $d_{1/2}$ such that

$$\begin{aligned} \partial _t d_{\frac{1}{2}}+\hat{\textbf{v}}_0\cdot \nabla d_{\frac{1}{2}}+\hat{\textbf{v}}_{\frac{1}{2}}\cdot \nabla d_{\Gamma }-\hat{\phi }_0d_{\frac{1}{2}}-\Delta d_{\Gamma }&=0\qquad \text {on } \Gamma , \end{aligned}$$

(3.31)

which will ensure that (3.21) is well-defined and we can choose $\hat{c}_{1}=0$, cf. (3.36) below. In what follows (see Corollary 3.7 below) we will find that $\hat{\textbf{v}}_0,\hat{\textbf{v}}_{\frac{1}{2}}\cdot \nabla d_{\Gamma }$ and $\hat{\phi }_0$ will be independent of $\rho$ on $\Gamma$.

To proceed we use that for $\mathbb {D}= \mathbb {A}, \mathbb {B}, \mathbb {C}$

$$\begin{aligned} \mathbb {D}_\varepsilon (\rho ,x,t)=\sum _{k\in \frac{1}{2}\mathbb {N}_0}\varepsilon ^{k}\mathbb {D}_{k}(\rho ,x,t)\quad \text {for all } \rho \in \mathbb {R}, (x,t)\in \Gamma (3\delta ) \end{aligned}$$

since $\hat{c}_\varepsilon$, $d_\varepsilon$ and their derivatives have a corresponding expansion.

Matching the $O(\varepsilon ^0)$-terms in the Allen–Cahn equation (3.19), we find

$$\begin{aligned} \hat{c}_{0}(\rho ,x,t)=\theta _0(\rho )\quad \text {for all }\rho \in \mathbb {R}, (x,t)\in \Gamma (3\delta ). \end{aligned}$$

(3.32)

Matching the $O(\varepsilon ^0)$-terms in the transformed momentum equation (3.17) and the divergence equation (3.18), we derive the following ordinary differential equations in $\rho$:

$$\begin{aligned} \partial _{\rho }\bigg (\nu (\theta _0)\big (\partial _{\rho }\hat{\textbf{v}}_0-\textbf{u}_0d_{\Gamma }\eta '\big )\bigg )&=\big (2\theta _0'\theta _0''+\partial _{\rho }\hat{p}_{-1}\big )\nabla d_{\Gamma }, \end{aligned}$$

(3.33)

$$\begin{aligned} \big (\partial _{\rho }\hat{\textbf{v}}_0-\textbf{u}_0d_{\Gamma }\eta '\big )\cdot \nabla d_{\Gamma }&=0, \end{aligned}$$

(3.34)

where the right-hand side vanishes for the choice $p_{-1}(\rho )= \frac{1}{2}\theta _0'(\rho )^2$. Matching the $O(\varepsilon ^{\frac{1}{2}})$-order terms in the Allen–Cahn equation (3.19), we find

$$\begin{aligned} \partial ^2_{\rho }\hat{c}_{\frac{1}{2}}-f''(\theta _0)\hat{c}_{\frac{1}{2}}=\theta _0'\big (\partial _t d_{\Gamma }+\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }-\hat{\phi }_0d_{\Gamma }\big )=0, \end{aligned}$$

which is compatible with the choice $\hat{c}_{\frac{1}{2}}=0$ above. Here we have used (3.20) and (3.25). Then matching the $O(\varepsilon ^{\frac{1}{2}})$-order terms in the transformed momentum equation and the divergence equation, we obtain the following ordinary differential equations with respect to $\rho$:

$$\begin{aligned} \partial _{\rho }\bigg (\nu (\theta _0)\big (\partial _{\rho }\hat{\textbf{v}}_{\frac{1}{2}}-(\textbf{u}_{\frac{1}{2}}d_{\Gamma }+\textbf{u}_0d_{\frac{1}{2}})\eta '\big )\bigg )&=\left( 2\theta _0'\theta _0''+\partial _{\rho }\hat{p}_{-1}\right) \nabla d_{\frac{1}{2}}+\partial _{\rho }\hat{p}_{-\frac{1}{2}}\nabla d_{\Gamma }, \end{aligned}$$

(3.35)

$$\begin{aligned} \big (\partial _{\rho }\hat{\textbf{v}}_{\frac{1}{2}}-(\textbf{u}_{\frac{1}{2}}d_{\Gamma }+\textbf{u}_0d_{\frac{1}{2}})\eta '\big )\cdot \nabla d_{\Gamma }&=\left( -\partial _{\rho }\hat{\textbf{v}}_{0}+\textbf{u}_{0}d_{\Gamma }\eta '\right) \cdot \nabla d_{\frac{1}{2}}. \end{aligned}$$

(3.36)

Furthermore comparing the $O(\varepsilon )$-order terms in the Allen–Cahn equation (3.19), we have

$$\begin{aligned} \partial ^2_{\rho }\hat{c}_{1}-f''(\theta _0)\hat{c}_{1}=\theta _0'\big (\partial _t d_{\frac{1}{2}}+\hat{\textbf{v}}_0\cdot \nabla d_{\frac{1}{2}}+\hat{\textbf{v}}_{\frac{1}{2}}\cdot \nabla d_{\Gamma }-\hat{\phi }_0d_{\frac{1}{2}}-\hat{\phi }_{\frac{1}{2}}d_{\Gamma }-\Delta d_{\Gamma }\big )=0, \end{aligned}$$

(3.37)

which again justifies the choice $\hat{c}_{1}=0$.

Then matching the $O(\varepsilon )$-order terms in the transformed momentum equation and the divergence equation, we obtain the following ordinary differential equations

$$\begin{aligned}&\partial _{\rho }\bigg (\nu (\theta _0)\big (\partial _{\rho }\hat{\textbf{v}}_{1}-(\textbf{u}_{1}d_{\Gamma }+\textbf{u}_{0}d_{1})\eta '\big )\bigg )=2\theta _0'\theta _0''\nabla d_{1}+\partial _{\rho }\hat{\textbf{v}}_0\partial _{t}d_{\Gamma }+\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }\partial _{\rho }\hat{\textbf{v}}_0\nonumber \\&\quad -\partial _{\rho }\big (2\nu (\theta _0)D\hat{\textbf{v}}_0\big )\cdot \nabla d_{\Gamma } -{\text {div}}\big (2\nu (\theta _0)D_d\hat{\textbf{v}}_0\big )+\partial _{\rho }\big (\nu (\theta _0){\text {div}}\hat{\textbf{v}}_0\big )\nabla d_{\Gamma }\nonumber \\&\quad +\sum \limits _{i\in \{-1,-\frac{1}{2},0\}}\partial _{\rho }\hat{p}_i\nabla d_{-i}+\mathbb {A}_0+\nabla \hat{p}_{-1} +\big (\nu (\theta _0)\eta '\big )'\big (\textbf{u}_{\frac{1}{2}}d_{\frac{1}{2}}-\textbf{u}_0\rho \big )+\textbf{l}_{0}d_{\Gamma }\eta ', \end{aligned}$$

(3.38)

and

$$\begin{aligned}&\big (\partial _{\rho }\hat{\textbf{v}}_{1}-(\textbf{u}_{1}d_{\Gamma }+\textbf{u}_{0}d_{1})\eta '\big )\cdot \nabla d_{\Gamma }=-\sum \limits _{i\in \{0,\frac{1}{2}\}}\partial _{\rho }\hat{\textbf{v}}_{i}\nabla d_{1-i}-{\text {div}}_x\hat{\textbf{v}}_0-\textbf{u}_0\cdot \nabla d_{\Gamma }\eta '\rho \nonumber \\ {}&\qquad +\eta '\sum \limits _{i\in \{\frac{1}{2},1\}}\textbf{u}_{0}\cdot \nabla d_{i}d_{1-i}+\eta '\sum \limits _{i\in \{0,\frac{1}{2}\}}\textbf{u}_{\frac{1}{2}}\cdot \nabla d_{i}d_{\frac{1}{2}-i}. \end{aligned}$$

(3.39)

Similarly, comparing the $O(\varepsilon ^{k})$-order terms for $k\ge \frac{3}{2}$ in the transformed momentum equation (3.17), the divergence equation (3.18) and the Allen–Cahn equation (3.19), we obtain the following ordinary differential equations

$$\begin{aligned}&\partial _{\rho }\left( \nu (\theta _0)\big (\partial _{\rho }\hat{\textbf{v}}_k-(\textbf{u}_kd_{\Gamma }+\textbf{u}_0d_{k})\eta '\big )\right) \nonumber \\&=-\sum \limits _{i\in \{k-1,k-\frac{1}{2},k\}}\partial _\rho \big (\hat{\nu }_i\partial _{\rho }\hat{\textbf{v}}_{k-i}\big ) +2\partial _{\rho }(\partial _{\rho }\hat{c}_k\theta _0')\nabla d_{\Gamma }+2\partial _{\rho }(\partial _{\rho }\hat{c}_{k-\frac{1}{2}}\theta _0')\nabla d_{\frac{1}{2}}+2\partial _{\rho }(\partial _{\rho }\hat{c}_{k-1}\theta _0')\nabla d_{1} \nonumber \\&\quad +\sum \limits _{i\in \{0,k-1\}}\partial _{\rho }\hat{\textbf{v}}_i\partial _td_{k-1-i}+\partial _{\rho }\hat{\textbf{v}}_0\sum \limits _{i\in \{0,k-1\}}\hat{\textbf{v}}_i\cdot \nabla d_{k-1-i}+\partial _{\rho }\hat{\textbf{v}}_{k-1}\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma } \nonumber \\&\quad +\theta _0''\nabla \hat{c}_{k-1}+(\theta _0')^2\big (\Delta d_{\Gamma }\nabla d_{k-1}+\Delta d_{k-1}\nabla d_{\Gamma }\big )+\eta '(\rho )\sum \limits _{i\in \{0,k-1\}}\textbf{l}_{i}d_{k-1-i} \nonumber \\&\quad -2\nabla d_{\Gamma }\cdot \sum \limits _{i\in \{0,k-1\}}\partial _\rho (\hat{\nu }_i D\hat{\textbf{v}}_{k-1-i})-2\nabla d_{k-1}\cdot \partial _\rho (\hat{\nu }_0 D\hat{\textbf{v}}_{0})-2{\text {div}}(\hat{\nu }_0 {(D_d\hat{\textbf{v}})_{k-1}})-2{\text {div}}(\hat{\nu }_{k-1} D_d\hat{\textbf{v}}_{0}) \nonumber \\&\quad +\nabla d_{\Gamma }\sum \limits _{i\in \{0,k-1\}}\partial _\rho (\hat{\nu }_i {\text {div}}\hat{\textbf{v}}_{k-1-i})+\nabla d_{k-1}\partial _\rho (\hat{\nu }_0 {\text {div}}\hat{\textbf{v}}_{0})+\sum \limits _{i\in \{0,k-1\}}\partial _\rho \hat{p}_i\nabla d_{k-1-i} \nonumber \\&\quad +\partial _\rho \big (\nu (\theta _0)\eta '\big )\sum \limits _{i\in \{\frac{1}{2},1,k-\frac{1}{2},k-1\}}\textbf{u}_{i}d_{k-i}-\partial _\rho \big (\nu (\theta _0)\eta '\big )\rho \textbf{u}_{k-1} +\mathcal {R}_{1,k-\frac{3}{2}}, \end{aligned}$$

(3.40)

$$\begin{aligned}&\big (\partial _{\rho }\hat{\textbf{v}}_k-\eta '(\textbf{u}_kd_{\Gamma }+\textbf{u}_0d_{k}))\cdot \nabla d_\Gamma \nonumber \\&\quad =-\sum \limits _{i\in \{0,\frac{1}{2},1,k-1,k-\frac{1}{2}\}}\partial _{\rho }\hat{\textbf{v}}_{i}\nabla d_{k-i}-{\text {div}}\hat{\textbf{v}}_{k-1}+\eta '\textbf{u}_0\cdot \sum \limits _{i\in \{\frac{1}{2},1,k-1,k-\frac{1}{2},k\}}\nabla d_{i}d_{k-i} \nonumber \\&\qquad +\eta '\textbf{u}_{\frac{1}{2}}\cdot \sum \limits _{i\in \{\frac{1}{2},k-1,k-\frac{1}{2}\}}\nabla d_{i}d_{k-\frac{1}{2}-i}+\eta '\textbf{u}_1\cdot \sum \limits _{i\in \{0,k-1\}}\nabla d_{i}d_{k-1-i} \nonumber \\&\qquad +\eta '\textbf{u}_{k-\frac{1}{2}}\cdot \sum \limits _{i\in \{0,\frac{1}{2}\}}\nabla d_{i}d_{\frac{1}{2}-i}+\eta '\textbf{u}_{k-1}\cdot \sum \limits _{i\in \{0,1\}}\nabla d_{i}d_{1-i} \nonumber \\&\qquad -\rho \eta '\sum \limits _{i\in \{0,k-1\}}\textbf{u}_i\cdot \nabla d_{k-1-i} +\mathcal {R}_{2,k-\frac{3}{2}}, \end{aligned}$$

(3.41)

and

$$\begin{aligned}&\partial ^2_{\rho }\hat{c}_{k}-f''(\theta _0)\hat{c}_{k}=\hat{f}'_{k-1}+\theta _0'\partial _t d_{k-\frac{1}{2}}+\partial _{\rho }\hat{c}_{k-\frac{1}{2}}\partial _t d_{\Gamma }+\partial _{\rho }\hat{c}_{k-1}\partial _t d_{\frac{1}{2}}\nonumber \\&\quad +\theta _0'\sum \limits _{i\in \{0,\frac{1}{2},k-1,k-\frac{1}{2}\}}\hat{\textbf{v}}_i\cdot \nabla d_{k-\frac{1}{2}-i} +\partial _{\rho }\hat{c}_{k-\frac{1}{2}}\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }+\partial _{\rho }\hat{c}_{k-1}\sum \limits _{i\in \{0,\frac{1}{2}\}}\hat{\textbf{v}}_i\cdot \nabla d_{\frac{1}{2}-i}\nonumber \\ {}&\quad -\sum \limits _{i\in \{0,k-1\}}\partial _{\rho }\hat{c}_i\Delta d_{k-1-i} - 2\nabla \partial _{\rho }\hat{c}_{k-1}\cdot \nabla d_{\Gamma }-\theta _0'\sum \limits _{i\in \{0,\frac{1}{2}\}}\hat{\phi }_id_{k-\frac{1}{2}-i} \nonumber \\ {}&\quad +\eta 'g_{k-\frac{3}{2}}d_{\Gamma }+\mathcal {R}_{3,k-\frac{3}{2}}, \end{aligned}$$

(3.42)

where

$$\begin{aligned} {\big (D_d\hat{\textbf{v}}\big )_{k-1}=\frac{1}{2}\big (\partial _\rho \hat{\textbf{v}}_{k-1}\otimes \nabla d_{\Gamma }+(\partial _\rho \hat{\textbf{v}}_{k-1}\otimes \nabla d_{\Gamma })^T\big )+\frac{1}{2}\big (\partial _\rho \hat{\textbf{v}}_0\otimes \nabla d_{k-1}+(\partial _\rho \hat{\textbf{v}}_0\otimes \nabla d_{k-1})^T\big )} \end{aligned}$$

and $\mathcal {R}_{1,k-\frac{3}{2}}$, $\mathcal {R}_{2,k-\frac{3}{2}}$, $\mathcal {R}_{3,k-\frac{3}{2}}$ depend on the terms up to $k-\frac{3}{2}$ order and converge exponentially to zero as $|\rho |\rightarrow \infty$ because of the choice of $W^\pm _k$ and ${\text {div}}\textbf{v}_k^\pm =0$ for all $k\in \frac{1}{2}\mathbb {N}_0$.

3.3 Existence of expansion terms

The following two lemmas are used to solve the ordinary differential equations with respect to $\rho$ and can be found in [3, Lemma A.2 and Lemma A.3] with f replaced by $f'$.

Lemma 3.3

Let $U\subset \mathbb {R}^n$ be an open subset and let $A:\mathbb {R}\times U\rightarrow \mathbb {R},(\rho ,x)\mapsto A(\rho ,x)$ be given and smooth. Assume that there exists $A^\pm (x)$ such that the decay property $A(\pm \rho ,x)-A^\pm (x)=O(e^{-\alpha \rho })$ as $\rho \rightarrow \infty$ is fulfilled. Then for every $x\in U$ the system

$$\begin{aligned} \begin{aligned}&w_{\rho \rho }(\rho ,x)-{f''}(\theta _0(\rho ))w(\rho ,x)=A(\rho ,x)\quad \text {for all } \rho \in \mathbb {R},\\&w(0,x)=0,\ w(\cdot ,x)\in L^\infty (\mathbb {R}) \end{aligned} \end{aligned}$$

(3.43)

has a smooth and bounded solution if and only if

$$\begin{aligned} \int _{\mathbb {R}}A(\rho ,x)\theta _0'(\rho )\,\textrm{d}\rho =0. \end{aligned}$$

(3.44)

In addition, if the solution exists, then it is unique and satisfies for all $x\in U$

$$\begin{aligned} \partial _\rho ^\ell \bigg (w(\pm \rho ,x)+\frac{A^\pm (x)}{f''(\pm 1)}\bigg )=O(e^{-\alpha \rho })\quad \text {as}~\rho \rightarrow \infty ,\ l=0,1,2. \end{aligned}$$

(3.45)

Furthermore, if $A(\rho ,x)$ satisfies for all $x\in U$

$$\begin{aligned} \partial _x^m\partial _\rho ^\ell \bigg (A(\pm \rho ,x)-A^\pm (x)\bigg )=O(e^{-\alpha \rho })\quad \text {as}~ \rho \rightarrow \infty \end{aligned}$$

for all $m\in \{0,\cdots ,M\}$ and $l\in \{0,\cdots ,L\}$, then

$$\begin{aligned} \partial _x^m\partial _\rho ^\ell \bigg (w(\pm \rho ,x)+\frac{A^\pm (x)}{f''(\pm 1)}\bigg )=O(e^{-\alpha \rho })\quad \text {as}~\rho \rightarrow \infty \end{aligned}$$

(3.46)

for all $m\in \{0,\cdots ,M\}$ and $l\in \{0,\cdots ,L\}$.

Lemma 3.4

Let $U\subset \mathbb {R}^n$ be an open subset and let $B:\mathbb {R}\times U\rightarrow \mathbb {R},(\rho ,x)\mapsto B(\rho ,x)$ be given and smooth. Assume that for all $x\in U$ the decay property $B(\pm \rho ,x)=O(e^{-\alpha \rho })$ as $\rho \rightarrow \infty$ is fulfilled. Then for each $x\in U$ the problem

$$\begin{aligned} \partial _{\rho }\big (\nu (\theta _0)\partial _\rho w(\rho ,x)\big )=B(\rho ,x)\quad \text {for all } \rho \in \mathbb {R} \end{aligned}$$

(3.47)

has a solution $w(\cdot ,x)\in C^2(\mathbb {R})\cap L^\infty (\mathbb {R})$ if and only if

$$\begin{aligned} \int _{\mathbb {R}}B(\rho ,x)\textrm{d}\rho =0. \end{aligned}$$

(3.48)

Furthermore, if $w_{*}(\rho ,x)$ is such a solution, then all the solutions can be written as

$$\begin{aligned} w(\rho ,x)=w_{*}(\rho ,x)+c(x),\ \rho \in \mathbb {R} \end{aligned}$$

(3.49)

where $c:U\rightarrow \mathbb {R}$ is an arbitrary function. In particular, if (3.47) holds,

$$\begin{aligned} w_{*}(\rho ,x)=\int _{0}^\rho \frac{1}{\nu (\theta _0)}\int _{-\infty }^r B(s,x)\textrm{d}s\,\textrm{d}r,\ \rho \in \mathbb {R} \end{aligned}$$

(3.50)

is a solution. Additionally, if (3.47) holds for all $x\in U$ and there exist $M,L\in \mathbb {N}$ such that

$$\begin{aligned} \partial _x^m\partial _\rho ^lB(\pm \rho ,x)=O(e^{-\alpha \rho }) \ \text {as}\ \rho \rightarrow +\infty \end{aligned}$$

(3.51)

for all $m\in \{0,\cdots ,M\}$ and $l\in \{0,\cdots ,L\}$, then there exists smooth functions $w^+(x)$ and $w^-(x)$ such that

$$\begin{aligned} {\partial }_x^m{\partial }_\rho ^l\big (w(\pm \rho ,x)-w^{\pm }(x)\big )=O(e^{-\alpha \rho }) \ \text {as} \ \rho \rightarrow +\infty \end{aligned}$$

(3.52)

for all $m\in \{0,\cdots ,M\}$ and $l\in \{0,\cdots ,L+2\}$.

Remark 3.5

We note the statements on the asymptotics “$O(e^{-\alpha \rho })$ as $\rho \rightarrow \infty$” in Lemma 3.3 and Lemma 3.4 hold true uniformly with respect to $x\in U$ provided $A(\pm \rho ,x)-A^\pm (x)=O(e^{-\alpha \rho })$ and $B(\pm \rho ,x)=O(e^{-\alpha \rho })$ as $\rho \rightarrow \infty$ hold true uniformly with respect to $x\in U$.

3.3.1 Solving lower order terms

Solving $\hat{p}_{-1}$, $\textbf{v}_0$, $\textbf{u}_0$, and $\textbf{v}_0^\pm$: Firstly, using the matching conditions (3.26) and (3.25) we get $\hat{c}_{0}=\theta _0$ satisfying (1.13). Moreover, it follows from (3.32) multiplied with $\nabla d_\Gamma$ and (3.33) that

$$\begin{aligned} 0=\partial _{\rho }\left( \nu (\theta _0)({\partial _\rho }\hat{\textbf{v}}_0-\textbf{u}_0d_{\Gamma }\eta '(\rho ))\cdot \nabla d_\Gamma \right) =2\theta _0'\theta _0''+\partial _{\rho }\hat{p}_{-1}{\quad \text { in }\mathbb {R}\times \Gamma (3\delta ),} \end{aligned}$$

where we used $\nabla =\nabla _x$ and $|\nabla d_\Gamma |^2\equiv 1$. Hence

$$\begin{aligned} \hat{p}_{-1}(\rho ,x,t)&=-(\theta '_0(\rho ))^2\quad \text {and} \nonumber \\ \hat{\textbf{v}}_0(\rho ,x,t)&=\overline{\textbf{v}}_0(x,t)+\textbf{u}_0(x,t)d_{\Gamma }\big (\eta (\rho )-\tfrac{1}{2}\big ) \end{aligned}$$

(3.53)

for all $\rho \in \mathbb {R}$, $(x,t)\in \Gamma (3\delta )$ and some function $\overline{\textbf{v}}_0:\Gamma (3\delta )\rightarrow \mathbb {R}^2$ due to Lemma 3.4. Using the matching conditions we obtain

$$\begin{aligned} \textbf{v}_{0}^+(x,t)=\overline{\textbf{v}}_0(x,t)+\frac{1}{2}\textbf{u}_0(x,t)d_{\Gamma }(x,t),\ \ \textbf{v}_{0}^-(x,t)=\overline{\textbf{v}}_0(x,t)-\frac{1}{2}\textbf{u}_0(x,t)d_{\Gamma }(x,t) \end{aligned}$$

and therefore

$$\begin{aligned} \overline{\textbf{v}}_0(x,t)=\frac{1}{2}\big (\textbf{v}_{0}^+(x,t)+\textbf{v}_{0}^-(x,t)\big ),\quad \textbf{u}_0(x,t)d_{\Gamma }(x,t)=\textbf{v}_{0}^+(x,t)-\textbf{v}_{0}^-(x,t) \end{aligned}$$

for all $(x,t)\in \Gamma (3\delta )$. The latter is consistent with (1.9) and yields

$$\begin{aligned} \hat{\textbf{v}}_0(\rho ,x,t)=\textbf{v}_{0}^+(x,t)\eta (\rho )+\textbf{v}_{0}^-(x,t)(1-\eta (\rho )), \end{aligned}$$

(3.54)

and

$$\begin{aligned} \textbf{u}_0(x,t)=\left\{ \begin{array}{ll} \frac{\textbf{v}_{0}^+(x,t)-\textbf{v}_{0}^-(x,t)}{d_{\Gamma }(x,t)}&{} \text {if }(x,t)\in \Gamma (3\delta )\backslash \Gamma ,\\ \\ \textbf{n}\cdot \nabla \big (\textbf{v}_{0}^+(x,t)-\textbf{v}_{0}^-(x,t)\big )&{} \text {if }(x,t)\in \Gamma . \end{array} \right. \end{aligned}$$

(3.55)

Therefore

$$\begin{aligned} \partial _t d_{\Gamma }+\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }=\partial _t d_{\Gamma }+\textbf{v}_{0}^-\cdot \nabla d_{\Gamma }+\big (\textbf{v}_{0}^+-\textbf{v}_{0}^-\big )\cdot \nabla d_{\Gamma }\eta (\rho ) \end{aligned}$$

(3.56)

which vanishes on $\Gamma$ due to (1.9) and (1.10).

Solving $\hat{p}_{-\frac{1}{2}}$and the $O(\varepsilon ^{\frac{1}{2}})$-order terms: It follows from (3.34) multiplied with $\nabla d_\Gamma$, (3.35), and (3.52) that

$$\begin{aligned} \partial _{\rho }\hat{p}_{-\frac{1}{2}}(\rho ,x,t)=0\qquad \text {for all }\rho \in \mathbb {R}, (x,t)\in \Gamma (3\delta ). \end{aligned}$$

Hence we can choose $\hat{p}_{-\frac{1}{2}}\equiv 0$. Using Lemma 3.4 we obtain

$$\begin{aligned} \hat{\textbf{v}}_{\frac{1}{2}}(\rho ,x,t)=\overline{\textbf{v}}_{\frac{1}{2}}(x,t)+\big (\textbf{u}_{\frac{1}{2}}(x,t)d_{\Gamma }(x,t)+\textbf{u}_{0}(x,t)d_{\frac{1}{2}}(x,t)\big )\big (\eta (\rho )-\tfrac{1}{2}\big ) \end{aligned}$$

for all $\rho \in \mathbb {R}$, $(x,t)\in \Gamma (3\delta )$ and some function $\overline{\textbf{v}}_{\frac{1}{2}}:\Gamma (3\delta )\rightarrow \mathbb {R}^2$. Because of the matching conditions we get

$$\begin{aligned} \textbf{v}_{\frac{1}{2}}^+=\overline{\textbf{v}}_{\frac{1}{2}}+\frac{1}{2}\big (\textbf{u}_{\frac{1}{2}}d_{\Gamma }+\textbf{u}_{0}d_{\frac{1}{2}}\big ),\ \ \textbf{v}_{\frac{1}{2}}^-=\overline{\textbf{v}}_{\frac{1}{2}}-\frac{1}{2}\big (\textbf{u}_{\frac{1}{2}}d_{\Gamma }+\textbf{u}_{0}d_{\frac{1}{2}}\big )\quad \text {in }\Gamma (3\delta ) \end{aligned}$$

as well as

$$\begin{aligned} \overline{\textbf{v}}_{\frac{1}{2}}=\frac{1}{2}\big (\textbf{v}_{\frac{1}{2}}^++\textbf{v}_{\frac{1}{2}}^-\big ),\ \ \textbf{u}_{\frac{1}{2}}d_{\Gamma }+\textbf{u}_{0}d_{\frac{1}{2}}=\textbf{v}_{\frac{1}{2}}^+-\textbf{v}_{\frac{1}{2}}^- \end{aligned}$$

(3.57)

which immediately imply

$$\begin{aligned} \hat{\textbf{v}}_{\frac{1}{2}}(\rho ,x,t)=\textbf{v}_{\frac{1}{2}}^+(x,t)\eta (\rho )+\textbf{v}_{\frac{1}{2}}^-(x,t)(1-\eta (\rho )) \end{aligned}$$

(3.58)

and

$$\begin{aligned} \textbf{u}_{\frac{1}{2}}(x,t)=\left\{ \begin{array}{ll} \frac{\textbf{v}_{\frac{1}{2}}^+(x,t)-\textbf{v}_{\frac{1}{2}}^-(x,t)-\textbf{u}_{0}(x,t)d_{\frac{1}{2}}(x,t)}{d_{\Gamma }(x,t)}&{}\text {if } (x,t)\in \Gamma (3\delta )\backslash \Gamma ,\\ \\ \textbf{n}\cdot \nabla \big (\textbf{v}_{\frac{1}{2}}^+(x,t)-\textbf{v}_{\frac{1}{2}}^-(x,t)-\textbf{u}_{0}(x,t)d_{\frac{1}{2}}(x,t)\big )&{}\text {if } (x,t)\in \Gamma \end{array} \right. \end{aligned}$$

(3.59)

for all $(x,t)\in \Gamma (3\delta )$, $\rho \in \mathbb {R}$.

Solving the $O(\varepsilon )$-order terms: To proceed we give the following proposition, which can be found in [3, Proposition A.5].

Proposition 3.6

There hold

$$\begin{aligned} \int _{-\infty }^{\infty }\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }\partial _{\rho }\hat{\textbf{v}}_{0}\textrm{d}\rho&=\frac{1}{2}\big (\textbf{v}_0^++\textbf{v}_0^-\big )\cdot \nabla d_{\Gamma }\big (\textbf{v}_0^+-\textbf{v}_0^-\big )=0&\quad&\text {on } \Gamma , \end{aligned}$$

(3.60)

$$\begin{aligned} \int _{-\infty }^{\infty }{\text {div}}\big (2\nu (\theta _0)D_d\hat{\textbf{v}}_0\big )\big )\textrm{d}\rho&=\overline{\nu }\widetilde{{\text {div}}}\hat{\textbf{v}}_0=\overline{\nu }\textbf{u}_0{} & {} \text {on } \Gamma ,\end{aligned}$$

(3.61)

$$\begin{aligned} \int _{-\infty }^{\infty }\mathbb {A}_0\textrm{d}\rho&=\sigma \Delta d_{\Gamma }\nabla d_{\Gamma }= -\sigma H \textbf{n}{} & {} \text {on }\Gamma ,\end{aligned}$$

(3.62)

$$\begin{aligned} \textbf{u}_0\cdot \textbf{n}&=0{} & {} \text {on } \Gamma , \end{aligned}$$

(3.63)

where $\widetilde{{\text {div}}}\hat{\textbf{v}}_0=\left( (\textbf{v}_{0}^+-\textbf{v}_{0}^-)\cdot \nabla \right) \nabla d_{\Gamma }+(\textbf{v}_{0}^+-\textbf{v}_{0}^-)\Delta d_{\Gamma }+(\nabla d_{\Gamma }\cdot \nabla )(\textbf{v}_{0}^+-\textbf{v}_{0}^-)$ and $\overline{\nu }$ is defined as in (3.14).

We need to point out that the first equalities in (3.59)–(3.61) hold not only on $\Gamma$ but also in $\Gamma (3\delta )$. Moreover, because of (1.9)

$$\begin{aligned} (\llbracket \textbf{v}_{0}^\pm \rrbracket \cdot \nabla )\nabla d_{\Gamma }=\big (\llbracket \textbf{v}_{0}^\pm \rrbracket \cdot \textbf{n}\big )\Delta d_{\Gamma }\textbf{n}=0,\quad \text {on }\Gamma . \end{aligned}$$

(3.64)

Corollary 3.7

$\hat{\textbf{v}}_0,\hat{\textbf{v}}_{\frac{1}{2}}\cdot \nabla d_{\Gamma }$ and $\hat{\phi }_0$ are independent of $\rho$ on $\Gamma$. Moreover, we can rewrite the evolution law (3.30) for $d_{1/2}$ as

$$\begin{aligned} \partial _t d_{\frac{1}{2}}+\textbf{v}_0^{\pm }\cdot \nabla d_{\frac{1}{2}}+\textbf{v}_{\frac{1}{2}}^{\pm }\cdot \nabla d_{\Gamma }&-\left( (\nabla d_{\Gamma }\cdot \nabla )\textbf{v}_{0}^{\pm }\cdot \nabla d_{\Gamma }\right) d_{\frac{1}{2}}-\Delta d_{\Gamma }=0\qquad \text {on } \Gamma . \end{aligned}$$

(3.65)

Proof

It follows from (3.53) that

$$\begin{aligned} \hat{\textbf{v}}_0(\rho ,x,t)=\textbf{v}_{0}^+(x,t)=\textbf{v}_{0}^-(x,t)\quad \text {on}\ \Gamma . \end{aligned}$$

(3.66)

By (3.56) and (3.57) one has

$$\begin{aligned} \hat{\textbf{v}}_{\frac{1}{2}}(\rho ,x,t)=\textbf{u}_{0}d_{\frac{1}{2}}\eta (\rho )+\textbf{v}_{\frac{1}{2}}^-(x,t)\quad \text {on}\ \Gamma . \end{aligned}$$

Together with (3.62) this leads to

$$\begin{aligned} \hat{\textbf{v}}_{\frac{1}{2}}(\rho ,x,t)\cdot \nabla d_{\Gamma }=\textbf{v}_{\frac{1}{2}}^-(x,t)\cdot \nabla d_{\Gamma }\quad \text {on}\ \Gamma . \end{aligned}$$

(3.67)

According to (3.23), (3.54) and (3.65) one has

$$\begin{aligned} \hat{\phi }_0&=\textbf{u}_{0}\cdot \nabla d_{\Gamma }\eta (\rho )+(\nabla d_{\Gamma }\cdot \nabla )\textbf{v}_{0}^-(x,t)\cdot \nabla d_{\Gamma }= (\nabla d_{\Gamma }\cdot \nabla )\textbf{v}_{0}^-(x,t)\cdot \nabla d_{\Gamma }\quad \text {on}\ \Gamma . \end{aligned}$$

(3.68)

This implies the statement. $\square$

Remark 3.8

We note that solvability of (3.64) together with a system for $\textbf{v}_{\frac{1}{2}}$ is given by Theorem A.14 in the appendix and will be discussed later.

In order to apply Lemma 3.4 to (3.37) the equation

$$\begin{aligned}&\big (p_0^+-p_0^-+\sigma \Delta d_{\Gamma }\big )\nabla d_{\Gamma }+\big (\textbf{v}_0^+-\textbf{v}_0^-\big )\partial _t d_{\Gamma }+\frac{1}{2}\big (\textbf{v}_0^++\textbf{v}_0^-\big )\cdot \nabla d_{\Gamma }\big (\textbf{v}_0^+-\textbf{v}_0^-\big )\nonumber \\ {}&\quad -2\big (\nu ^+ D\textbf{v}_0^+-\nu ^- D\textbf{v}_0^-\big )\cdot \nabla d_{\Gamma }-\overline{\nu }\widetilde{{\text {div}}}\hat{\textbf{v}}_0+\textbf{l}_{0}d_{\Gamma }+\overline{\nu }\textbf{u}_0=0 \end{aligned}$$

(3.69)

has to be satisfied on $\Gamma (3\delta )$ because of Proposition 3.6. Using (3.59)–(3.61) we obtain that (3.68) on $\Gamma$ is equivalent to

$$\begin{aligned} 2\llbracket \nu ^\pm D\textbf{v}_0^\pm \rrbracket \textbf{n}-\llbracket p_0^\pm \rrbracket \textbf{n}=\sigma \Delta d_{\Gamma }\textbf{n},\ \ \text {on}\ \Gamma , \end{aligned}$$

(3.70)

i.e., the balance of normal stresses (1.8) has to hold, which is true by our assumptions. In order to obtain (3.68) on $\Gamma (3\delta )$ we define

$$\begin{aligned} \textbf{l}_{0}=\left\{ \begin{array}{ll} \frac{\big (p_0^--p_0^+-\sigma \Delta d_{\Gamma }\big )\nabla d_{\Gamma }+\big (\textbf{v}_0^--\textbf{v}_0^+\big )\partial _t d_{\Gamma }-\frac{1}{2}\big (\textbf{v}_0^++\textbf{v}_0^-\big )\cdot \nabla d_{\Gamma }\big (\textbf{v}_0^+-\textbf{v}_0^-\big )}{d_{\Gamma }}\\ \qquad +\frac{2\big (\nu ^+ D\textbf{v}_0^+-\nu ^- D\textbf{v}_0^-\big )\cdot \nabla d_{\Gamma }+\overline{\nu }\big (\widetilde{{\text {div}}}\hat{\textbf{v}}_0-\textbf{u}_0\big )}{d_{\Gamma }}&{} \text {in }\Gamma (3\delta )\backslash \Gamma ,\\ \\ \textbf{n}\cdot \nabla \bigg (\big (p_0^--p_0^+-\sigma \Delta d_{\Gamma }\big )\nabla d_{\Gamma }+\big (\textbf{v}_0^--\textbf{v}_0^+\big )\partial _t d_{\Gamma }\\ \qquad -\frac{1}{2}\big (\textbf{v}_0^++\textbf{v}_0^-\big )\cdot \nabla d_{\Gamma }\big (\textbf{v}_0^+-\textbf{v}_0^-\big )+2\big (\nu ^+ D\textbf{v}_0^+-\nu ^- D\textbf{v}_0^-\big )\cdot \nabla d_{\Gamma }\\ \qquad +\overline{\nu }\big (\widetilde{{\text {div}}}\hat{\textbf{v}}_0-\textbf{u}_0\big )\bigg )&{} \text {on }\Gamma . \end{array} \right. \end{aligned}$$

(3.71)

Then the solution of (3.37) is given by

$$\begin{aligned} \hat{\textbf{v}}_1-(\textbf{u}_0d_{1}+\textbf{u}_1d_{\Gamma })\eta (\rho )&=\textbf{v}_1^-+\textbf{u}_{\frac{1}{2}}d_{\frac{1}{2}}\eta (\rho )+\int _{-\infty }^\rho \frac{1}{\nu (\theta _0(r))}\int _{-\infty }^r\textbf{V}^{0}(s,x,t)\,\textrm{d}s\,\textrm{d}r\nonumber \\&=:\textbf{v}_1^-+\textbf{W}^{0}\qquad \text {in }\mathbb {R}\times \Gamma (3\delta ) \end{aligned}$$

(3.72)

because of the matching conditions, where $\textbf{V}^{0}$ consists of the terms up to zero order. Passing to the limit $\rho \rightarrow \infty$ yields

$$\begin{aligned} \textbf{u}_0d_{1}+\textbf{u}_1d_{\Gamma }+\textbf{u}_{\frac{1}{2}}d_{\frac{1}{2}}+\textbf{W}^{0}|_{\rho =+\infty }=\textbf{v}_{1}^+-\textbf{v}_{1}^-, \end{aligned}$$

(3.73)

which immediately implies that

$$\begin{aligned} \hat{\textbf{v}}_{1}(\rho ,x,t)=\textbf{v}_{1}^+(x,t)\eta (\rho )+\textbf{v}_{1}^-(x,t)(1-\eta (\rho ))+\textbf{W}^{0}(\rho ,x,t)-\textbf{W}^{0}(x,t,+\infty )\eta (\rho ) \end{aligned}$$

(3.74)

for all $\rho \in \mathbb {R}$, $(x,t)\in \Gamma (3\delta )$ and

$$\begin{aligned} \textbf{u}_{1}=\left\{ \begin{array}{ll} \frac{\textbf{v}_{1}^+-\textbf{v}_{1}^--\textbf{u}_{0}d_{1}-\textbf{u}_{\frac{1}{2}}d_{\frac{1}{2}}-\textbf{W}^{0}|_{\rho =+\infty }}{d_{\Gamma }}&{} \text {in }\Gamma (3\delta )\backslash \Gamma ,\\ \\ \textbf{n}\cdot \nabla \big (\textbf{v}_{1}^+-\textbf{v}_{1}^--\textbf{u}_{0}d_{1}-\textbf{u}_{\frac{1}{2}}d_{\frac{1}{2}}-\textbf{W}^{0}|_{\rho =+\infty }\big )&{}\text {on } \Gamma . \end{array} \right. \end{aligned}$$

(3.75)

In order to determine $\hat{p}_0$ we multiply (3.37) with $\nabla d_\Gamma$, use (3.38), and obtain

$$\begin{aligned}&\partial _{\rho }\bigg (\hat{p}_0+\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }\partial _t d_{\Gamma }+\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }\hat{\textbf{v}}_{0}\cdot \nabla d_{\Gamma } -\big (2\nu (\theta _0 )D\hat{\textbf{v}}_0:\nabla d_{\Gamma }\otimes \nabla d_{\Gamma }\big )\nonumber \\ {}&\qquad -\int _{-\infty }^\rho {\text {div}}\big (2\nu (\theta _0(r))D_d\hat{\textbf{v}}_0(r,\cdot )\big )\cdot \nabla d_{\Gamma }\,\textrm{d}r +\nu (\theta _0){\text {div}}\hat{\textbf{v}}_0\nonumber \\ {}&\qquad +\textbf{l}_0d_{\Gamma }\eta \cdot \nabla d_{\Gamma } -\nu (\theta _0)\mathcal {A}^0+\int _{-\infty }^\rho \widetilde{\textbf{V}}^0(r,\cdot )\cdot \nabla d_{\Gamma }\textrm{d}r\bigg )=0, \end{aligned}$$

(3.76)

where $\mathcal {A}^{k-1}$ and $\widetilde{\textbf{V}}^{k-1}$ consist of some terms up to 0 order. Integrating this equation on $(-\infty ,\rho )$ and using the matching condition for $\hat{p}_0$ determines $\hat{p}_0$.

In summary the equations for $(\hat{\textbf{v}}_1, \textbf{u}_1, \hat{p}_0)$ are solvable if $(\textbf{v}_0^\pm , p_0^\pm , (\Gamma _t)_{t\in [0,T_0]})$ solves the sharp interface limit system (1.6)–(1.10) and we have:

Lemma 3.9

(The zeroth order terms)  

Let $\hat{p}_{-1}= (\theta _0')^2$, let $(\textbf{v}_0^\pm , p_0^\pm , (\Gamma _t)_{t\in [0,T_0]})$ be the solution of (1.6)–(1.10), and let $(\textbf{v}_0^{\pm },p_0^{\pm })$ be extended to $\Omega ^{\pm }\cup \Gamma (3\delta )$ as in Remark 3.1. Moreover, we define $c_{0}^{\pm }(x,t)=\pm 1$ for all $(x,t)\in \Omega ^{\pm }\cup \Gamma (3\delta )$ and $\hat{c}_{0},\hat{\textbf{v}}_{0}, \textbf{u}_0, \textbf{l}_0$ by (3.31), (3.53), (3.54), and (3.70), respectively. Then the outer equations (3.4), (3.5), (3.6) (for $k=0$), the inner equations (3.39), (3.40), (3.41) (for $k=0$), the inner–outer matching conditions (3.26) (for $k=0$) are satisfied on $\Gamma (3\delta )$. Finally, the compatibility condition for (3.37), which is equivalent to (3.68), is satisfied on $\Gamma (3\delta )$.

3.3.2 Solving the higher order terms

Determining $\hat{c}_{\frac{3}{2}}$, the evolution law of $d_1$ on $\Gamma$, and $g_0$: Taking $k=\frac{3}{2}$ in (3.41) one has

$$\begin{aligned} \partial ^2_{\rho }\hat{c}_{\frac{3}{2}}-f''(\theta _0)\hat{c}_{\frac{3}{2}}&=\theta _0'(\rho )\partial _t d_{1}+\theta _0'(\rho )\big (\hat{\textbf{v}}_0\cdot \nabla d_{1}+\hat{\textbf{v}}_{\frac{1}{2}}\cdot \nabla d_{\frac{1}{2}}+\hat{\textbf{v}}_1\cdot \nabla d_{\Gamma }\big )-\theta _0'(\rho )\Delta d_{\frac{1}{2}} \nonumber \\ {}&\quad -\theta _0'(\rho )\big (\hat{\phi }_0d_{1}+\hat{\phi }_{\frac{1}{2}}d_{\frac{1}{2}}\big )+\eta '(\rho )g_0d_{\Gamma }+\theta _0'(\rho )\rho \hat{\phi }_{0}. \end{aligned}$$

(3.77)

The compatibility condition (3.43) for (3.76) is equivalent to

$$\begin{aligned} \mathcal {D}_1+\sigma ^{-1}\sigma _1g_0d_{\Gamma }=0\qquad \text {in }\Gamma (3\delta ), \end{aligned}$$

(3.78)

where $\sigma _1:=\int _{-\infty }^{\infty }\theta _0'(\rho )\eta '(\rho )\textrm{d}\rho$

$$\begin{aligned} \mathcal {D}_1:=\partial _t d_{1}+\frac{\textbf{v}_0^{+}+\textbf{v}_0^{-}}{2}\cdot \nabla d_{1}+\frac{\textbf{v}_1^{+}+\textbf{v}_1^{-}}{2}\cdot \nabla d_{\Gamma }-\sigma ^{-1}\int _{-\infty }^{+\infty }\big (\theta _0'(\rho )\big )^2\hat{\phi }_0(\rho ,\cdot )\, \textrm{d}\rho \, d_{1}+D_{\frac{1}{2}} \end{aligned}$$

and $D_{\frac{1}{2}}$ depends on the terms up to order $\frac{1}{2}$, which were determined before. Here we have used the definition of $\hat{\textbf{v}}_0$ and $\int _{\mathbb {R}}\theta '(\rho )(\eta (\rho )-\tfrac{1}{2})\, \textrm{d}\rho =0$ since $\eta -\tfrac{1}{2}$ is odd. On $\Gamma$ the latter equation is satisfied if and only if $d_1$ solves the evolution equation

$$\begin{aligned} \partial _t d_{1}+\textbf{v}_0^{\pm }\cdot \nabla d_{1}+\frac{\textbf{v}_1^{+}+\textbf{v}_1^{-}}{2}\cdot \nabla d_{\Gamma }-\hat{\phi }_0 d_{1}+D_{\frac{1}{2}}=0\ \ \text {on}\ \Gamma . \end{aligned}$$

(3.79)

In order to satisfy the compatibility condition on $\Gamma (3\delta )\setminus \Gamma$ we define

$$\begin{aligned} g_0=\left\{ \begin{array}{ll} -\frac{\sigma \sigma _1^{-1}\mathcal {D}_1}{d_{\Gamma }}&{} \text {in }\Gamma (3\delta )\backslash \Gamma ,\\ \\ -\sigma \sigma _1^{-1}\textbf{n}\cdot \nabla \mathcal {D}_1&{} \text {on } \Gamma . \end{array} \right. \end{aligned}$$

(3.80)

Determining $\hat{c}_{k}$ for $k \ge 2$, the evolution law of $d_{k-\frac{1}{2}}$ on $\Gamma$, and $g_{k-\frac{3}{2}}$: First of all, we rewrite (3.41) as

$$\begin{aligned}&\partial ^2_{\rho }\hat{c}_{k}-f''(\theta _0)\hat{c}_{k}=\mathcal {S}_{1,k-\frac{3}{2}}(\theta _0,\hat{c}_{\frac{3}{2}},\cdots ,\hat{c}_{k-\frac{1}{2}},d_0,\cdots ,d_{k-1},\hat{\textbf{v}}_0,\cdots ,\hat{\textbf{v}}_{k-\frac{3}{2}},g_0,\cdots ,g_{k-2},\hat{\phi }_0,\hat{\phi }_{\frac{1}{2}}) \nonumber \\ {}&\quad +\theta _0'\partial _t d_{k-\frac{1}{2}}+\theta _0'\sum \limits _{j=0,j\in \frac{1}{2}\mathbb {N}_0}^{k-\frac{1}{2}}\hat{\textbf{v}}_j\cdot \nabla d_{k-\frac{1}{2}-j}-\theta _0'\Delta d_{k-1}-\theta _0'\hat{\phi }_0d_{k-\frac{1}{2}}+\eta 'g_{k-\frac{3}{2}}d_{0}, \end{aligned}$$

(3.81)

where $\mathcal {S}_{1,k-\frac{3}{2}}$ depends on lower order terms which are known by the induction hypothesis. Then the compatibility condition (3.43) for (3.80) is equivalent to

$$\begin{aligned} \mathcal {D}_{k-\frac{1}{2}}+\sigma ^{-1}\sigma _1g_{k-\frac{3}{2}}d_{\Gamma }=0\qquad \text {on }\Gamma (3\delta ), \end{aligned}$$

(3.82)

where

$$\begin{aligned} \mathcal {D}_{k-\frac{1}{2}}&=\sigma ^{-1}\int _{-\infty }^{+\infty }\theta _0'(\rho )\mathcal {S}_{1,k-\frac{3}{2}}\, \textrm{d}\rho +\partial _t d_{k-\frac{1}{2}}-\Delta d_{k-1} \nonumber \\ {}&\quad +\sigma ^{-1}\int _{-\infty }^{+\infty }\big (\theta _0'(\rho )\big )^2\bigg (\sum \limits _{j=0,j\in \frac{1}{2}\mathbb {N}_0}^{k-\frac{1}{2}}\hat{\textbf{v}}_j\cdot \nabla d_{k-\frac{1}{2}-j}-\hat{\phi }_0d_{k-\frac{1}{2}}\bigg )\textrm{d}\rho . \end{aligned}$$

It is satisfied if $d_{k-\frac{1}{2}}$ solves

$$\begin{aligned}&\partial _t d_{k-\frac{1}{2}}+\sigma ^{-1}\int _{-\infty }^{+\infty }\big (\theta _0'(\rho )\big )^2\bigg (\sum \limits _{j=0,j\in \frac{1}{2}\mathbb {N}_0}^{k-\frac{1}{2}}\hat{\textbf{v}}_j\cdot \nabla d_{k-\frac{1}{2}-j}-\hat{\phi }_0d_{k-\frac{1}{2}}\bigg )\textrm{d}\rho \nonumber \\ {}&-\Delta d_{k-1}+\sigma ^{-1}\int _{-\infty }^{+\infty }\theta _0'(\rho )\mathcal {S}_{1,k-\frac{3}{2}}\textrm{d}\rho =0\quad \text {on }\Gamma \end{aligned}$$

(3.83)

and

$$\begin{aligned} g_{k-\frac{3}{2}}=\left\{ \begin{array}{ll} -\frac{\sigma \sigma _1^{-1}\mathcal {D}_{k-\frac{1}{2}}}{d_{\Gamma }}&{} \text {in } \Gamma (3\delta )\backslash \Gamma ,\\ \\ -\sigma \sigma _1^{-1}\textbf{n}\cdot \nabla \mathcal {D}_{k-\frac{1}{2}}&{} \text {on }\Gamma . \end{array} \right. \end{aligned}$$

(3.84)

Determining $\hat{\textbf{v}}_{k}$, $k\ge \frac{3}{2}$, the jump conditions on $\Gamma$ and $\textbf{l}_{k-1}$: Firstly, the compatibility condition (3.47) for (3.39) is equivalent to

$$\begin{aligned}&\big (\textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\partial _td_{k-1}+\sigma \Delta d_{k-1}\nabla d_\Gamma +\big (\textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\frac{\textbf{v}_{0}^++\textbf{v}_{0}^-}{2}\cdot \nabla d_{k-1} \nonumber \\&\quad +\big (\sigma \Delta d_\Gamma +p_{0}^+- p_{0}^--2\big (\nu ^+ D\textbf{v}_{0}^+- 2\nu ^- D\textbf{v}_{0}^-\big )\big )\nabla d_{k-1} +\big (\textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\int _{-\infty }^{\infty }\eta '\hat{\textbf{v}}_{k-1}\cdot \nabla d_{\Gamma }\textrm{d}\rho \nonumber \\&\quad +\int _{-\infty }^{\infty }\partial _{\rho }\hat{\textbf{v}}_{k-1}\big (\partial _td_{\Gamma }+\hat{\textbf{v}}_0\cdot \nabla d_{\Gamma }\big )\textrm{d}\rho -\int _{-\infty }^{\infty }{\text {div}}\big (2\nu (\theta _0){\big (D_d\hat{\textbf{v}}\big )_{k-1}}\big )\big )\textrm{d}\rho \nonumber \\&\quad -\big ( 2\nu ^+ D\textbf{v}_{k-1}^+- 2\nu ^- D\textbf{v}_{k-1}^-\big )\nabla d_{\Gamma }+(p_{k-1}^+-p_{k-1}^-)\nabla d_{\Gamma }+\textbf{l}_{0}d_{k-1}+\textbf{l}_{k-1}d_{\Gamma }\nonumber \\&=\mathcal {S}_{2,k-\frac{3}{2}}(\theta _0,\hat{c}_{\frac{3}{2}},\cdots ,\hat{c}_{k-1},d_0,\cdots ,d_{k-2},\hat{\textbf{v}}_0,\cdots ,\hat{\textbf{v}}_{k-\frac{3}{2}},\textbf{l}_0,\cdots ,\textbf{l}_{k-\frac{3}{2}}). \end{aligned}$$

(3.85)

Here $\mathcal {S}_{2,k-\frac{3}{2}}$ depends on low order terms, which were determined before.

If it is satisfied, the solution to (3.39) is given by

$$\begin{aligned} \hat{\textbf{v}}_k-(\textbf{u}_0d_{k}+\textbf{u}_kd_{\Gamma })\eta&=\textbf{v}_k^-+(\textbf{u}_{\frac{1}{2}}d_{k-\frac{1}{2}}+\textbf{u}_{k-\frac{1}{2}}d_{\frac{1}{2}})\eta +\int _{-\infty }^\rho \frac{1}{\nu (\theta _0(r))}\int _{-\infty }^r\textbf{V}^{k-1}(s,x,t)\,\textrm{d}s\,\textrm{d}r\nonumber \\&=:\textbf{v}_k^-+(\textbf{u}_{\frac{1}{2}}d_{k-\frac{1}{2}}+\textbf{u}_{k-\frac{1}{2}}d_{\frac{1}{2}})\eta +\textbf{W}^{k-1}, \end{aligned}$$

(3.86)

where $\textbf{V}^{k-1}$ consists of terms up to $k-1$ order. By taking $\rho \rightarrow +\infty$ in (3.85) and the matching conditions for $\hat{\textbf{v}}_k$ we obtain

$$\begin{aligned} \textbf{v}_k^+-\textbf{v}_k^-=\textbf{u}_0d_{k}+\textbf{u}_kd_{\Gamma }+\textbf{u}_{\frac{1}{2}}d_{k-\frac{1}{2}}+\textbf{u}_{k-\frac{1}{2}}d_{\frac{1}{2}}+\textbf{W}^{k-1}|_{\rho =+\infty }, \end{aligned}$$

which yields

$$\begin{aligned} {\llbracket \textbf{v}_k^{\pm }\rrbracket \cdot \textbf{n}}&=\big (\textbf{u}_{\frac{1}{2}}d_{k-\frac{1}{2}}+\textbf{u}_{k-\frac{1}{2}}d_{\frac{1}{2}}+\textbf{W}^{k-1}|_{\rho =+\infty }\big )\cdot \textbf{n}+\textbf{u}_0\cdot \textbf{n}d_{k}=:\hat{a}_{k-\frac{1}{2}}\quad \text {on}\ \Gamma , \end{aligned}$$

(3.87)

where we have used $\textbf{u}_0\cdot \textbf{n}|_{\Gamma }=0$ due to Proposition 3.6, and

$$\begin{aligned} \llbracket \textbf{v}_k^{\pm }\rrbracket \cdot {\varvec{\tau }}&=\big (\textbf{u}_{\frac{1}{2}}d_{k-\frac{1}{2}}+\textbf{u}_{k-\frac{1}{2}}d_{\frac{1}{2}}+\textbf{W}^{k-1}|_{\rho =+\infty }\big )\cdot {\varvec{\tau }}+\textbf{u}_0\cdot {\varvec{\tau }}d_{k} =:\check{a}_{k-\frac{1}{2}}+\textbf{u}_0\cdot {\varvec{\tau }}d_{k} \end{aligned}$$

(3.88)

on $\Gamma$. Since by the induction hypothesis one assumes that the compatibility condition (3.84) for $k-1$ instead of k is already satisfied, one obtains

$$\begin{aligned} \hat{\textbf{v}}_{k-1}&=\textbf{v}_{k-1}^+\eta (\rho )+\textbf{v}_{k-1}^-(1-\eta (\rho ))+(1-\eta (\rho ))\int _{-\infty }^\rho \frac{1}{\nu (\theta _0(r))}\int _{-\infty }^r\textbf{V}^{k-2}(s,\cdot )\,\textrm{d}s\,\textrm{d}r \nonumber \\ {}&\quad -\eta (\rho )\int _\rho ^{+\infty }\frac{1}{\nu (\theta _0(r))}\int _{-\infty }^r\textbf{V}^{k-2}(s,\cdot )\,\textrm{d}s\,\textrm{d}r, \end{aligned}$$

(3.89)

where $\textbf{V}_{-1}\equiv 0$. Inserting this we can rewrite (3.84) as

$$\begin{aligned} \mathcal {J}_{k-1}+\textbf{l}_{k-1}d_{\Gamma }=\tilde{\mathcal {S}}_{k-\frac{3}{2}}, \end{aligned}$$

(3.90)

where

$$\begin{aligned} \mathcal {J}_{k-1}&=\big (\textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\partial _td_{k-1}+\sigma \Delta d_{k-1}\nabla d_\Gamma +\big (\textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\frac{\textbf{v}_{0}^++\textbf{v}_{0}^-}{2}\cdot \nabla d_{k-1} \nonumber \\&\quad +\big (\sigma \Delta d_\Gamma +p_{0}^+- p_{0}^--2\big (\nu ^+ D\textbf{v}_{0}^+- 2\nu ^- D\textbf{v}_{0}^-\big )\big )\nabla d_{k-1}+\textbf{l}_{0}d_{k-1} \nonumber \\&\quad +\big (\textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\frac{\textbf{v}_{k-1}^++\textbf{v}_{k-1}^-}{2}\cdot \nabla d_{\Gamma }+ \big (\textbf{v}_{k-1}^+- \textbf{v}_{k-1}^-\big )\partial _td_{\Gamma }+\big (\textbf{v}_{k-1}^+- \textbf{v}_{k-1}^-\big )\frac{\textbf{v}_{0}^++\textbf{v}_{0}^-}{2}\cdot \nabla d_{\Gamma } \nonumber \\ {}&\quad -\overline{\nu }\left( ( \textbf{v}_{k-1}^+- \textbf{v}_{k-1}^-)\cdot \nabla \right) \nabla d_{\Gamma }-\overline{\nu }\big ( \textbf{v}_{k-1}^+- \textbf{v}_{k-1}^-\big )\Delta d_{\Gamma }-\overline{\nu }(\nabla d_{\Gamma }\cdot \nabla )\big ( \textbf{v}_{k-1}^+- \textbf{v}_{k-1}^-\big ) \nonumber \\ {}&\quad {-\overline{\nu }\left( ( \textbf{v}_{0}^+- \textbf{v}_{0}^-)\cdot \nabla \right) \nabla d_{k-1}-\overline{\nu }\big ( \textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\Delta d_{k-1}-\overline{\nu }\nabla \big ( \textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\nabla d_{k-1}} \nonumber \\ {}&\quad -\big ( 2\nu ^+ D\textbf{v}_{k-1}^+- 2\nu ^- D\textbf{v}_{k-1}^-\big )\nabla d_{\Gamma }+(p_{k-1}^+-p_{k-1}^-)\nabla d_{\Gamma }. \end{aligned}$$

(3.91)

Here we have used

$$\begin{aligned} \int _{-\infty }^{\infty }{\text {div}}\big (2\nu (\theta _0)\big (D_d\hat{\textbf{v}}\big )_{k-1}\big )\big )\textrm{d}\rho&=\overline{\nu }\left( ( \textbf{v}_{k-1}^+- \textbf{v}_{k-1}^-)\cdot \nabla \right) \nabla d_{\Gamma }+\overline{\nu }\big ( \textbf{v}_{k-1}^+- \textbf{v}_{k-1}^-\big )\Delta d_{\Gamma }\\ {}&\quad +\overline{\nu }(\nabla d_{\Gamma }\cdot \nabla )\big ( \textbf{v}_{k-1}^+- \textbf{v}_{k-1}^-\big ) +\overline{\nu }\left( ( \textbf{v}_{0}^+- \textbf{v}_{0}^-)\cdot \nabla \right) \nabla d_{k-1} \\ {}&\quad +\overline{\nu }\big ( \textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\Delta d_{k-1}+\overline{\nu }\nabla \big ( \textbf{v}_{0}^+- \textbf{v}_{0}^-\big )\nabla d_{k-1}, \end{aligned}$$

which is shown in the same way as (3.60).

Therefore (3.84) is satisfied if

$$\begin{aligned} \tilde{\mathcal {S}}_{k-\frac{3}{2}}&=\sigma \Delta d_{k-1}\nabla d_\Gamma +\big (\sigma \Delta d_\Gamma +\llbracket p_{0}^\pm \rrbracket -2\llbracket \nu ^\pm D\textbf{v}_{0}^\pm \rrbracket {-\overline{\nu }\llbracket \nabla \textbf{v}_{0}^\pm \rrbracket \big )\nabla d_{k-1}}+\textbf{l}_{0}d_{k-1} \nonumber \\&\quad + \llbracket \textbf{v}_{k-1}^\pm \rrbracket \big (\partial _td_{\Gamma }+\textbf{v}_{0}^\pm \cdot \nabla d_{\Gamma }-\overline{\nu }\Delta d_{\Gamma }-\overline{\nu }\nabla ^2d_{\Gamma }\big ) \nonumber \\ {}&-\overline{\nu }\llbracket \nabla \textbf{v}_{k-1}^\pm \rrbracket \cdot \nabla d_{\Gamma }-2\llbracket \nu ^\pm D\textbf{v}_{k-1}^\pm \rrbracket \nabla d_{\Gamma }+\llbracket p_{k-1}^\pm \rrbracket \nabla d_{\Gamma } \quad \text {on }\Gamma \end{aligned}$$

(3.92)

and

$$\begin{aligned} \textbf{l}_{k-1}=\left\{ \begin{array}{ll} \frac{\tilde{\mathcal {S}}_{k-\frac{3}{2}}-\mathcal {J}_{k-1}}{d_{\Gamma }}&{} \text {in } \Gamma (3\delta )\backslash \Gamma ,\\ \\ \textbf{n}\cdot \nabla \big (\tilde{\mathcal {S}}_{k-\frac{3}{2}}-\mathcal {J}_{k-1}\big )&{} \text {on } \Gamma . \end{array} \right. \end{aligned}$$

(3.93)

In order to satisfy the matching conditions for $\hat{\textbf{v}}_k$ we define $\textbf{u}_{k}$ by

$$\begin{aligned} \textbf{u}_k=\left\{ \begin{array}{ll} \frac{\textbf{v}_k^{+}-\textbf{v}_k^{-}-\textbf{u}_{0}d_{k}-\textbf{u}_{\frac{1}{2}}d_{k-\frac{1}{2}}-\textbf{u}_{k-\frac{1}{2}}d_{\frac{1}{2}}-\textbf{W}^{k-1}|_{\rho =+\infty }}{d_{\Gamma }}&{}\ \text {on } \Gamma (3\delta )\setminus \Gamma ,\\ \textbf{n}\cdot \nabla \big (\textbf{v}_k^{+}-\textbf{v}_k^{-}-\textbf{u}_{0}d_{k}-\textbf{u}_{\frac{1}{2}}d_{k-\frac{1}{2}}-\textbf{u}_{k-\frac{1}{2}}d_{\frac{1}{2}}-\textbf{W}^{k-1}|_{\rho =+\infty }\big )&{}\ \text {on } \Gamma . \end{array} \right. \end{aligned}$$

(3.94)

Then

$$\begin{aligned} \hat{\textbf{v}}_k(\rho ,x,t)&=\textbf{v}_k^+(x,t)\eta (\rho )+\textbf{v}_k^-(x,t)(1-\eta (\rho ))+(1-\eta (\rho ))\int _{-\infty }^\rho \frac{1}{\nu (\theta _0(r))}\int _{-\infty }^r\textbf{V}^{k-1}(s,x,t)\,\textrm{d}s\,\textrm{d}r \nonumber \\ {}&\quad -\eta \int _\rho ^{+\infty }\frac{1}{\nu (\theta _0(r))}\int _{-\infty }^r\textbf{V}^{k-1}(s,x,t)\,\textrm{d}s\,\textrm{d}r, \end{aligned}$$

(3.95)

satisfies the inner–outer matching conditions (3.26).

In order to determine $\hat{p}_{k-1}$ we multiply (3.39) by $\nabla d_{\Gamma }$ and use (3.40). This yields

$$\begin{aligned}&\partial _{\rho }\bigg (\hat{p}_{k-1}+\hat{\textbf{v}}_{k-1}\cdot \nabla d_{\Gamma }\partial _t d_{\Gamma }+\hat{\textbf{v}}_{k-1}\cdot \nabla d_{\Gamma }\hat{\textbf{v}}_{0}\cdot \nabla d_{\Gamma } -\big (2\nu (\theta _0 )D\hat{\textbf{v}}_{k-1}:\nabla d_{\Gamma }\otimes \nabla d_{\Gamma }\big )\nonumber \\ {}&\qquad -\int _{-\infty }^\rho {\text {div}}\big (2\nu (\theta _0(r)){\big (D_d\hat{\textbf{v}}\big )_{k-1}}\big )(r,\cdot )\big )\cdot \nabla d_{\Gamma }\,\textrm{d}r +\nu (\theta _0){\text {div}}\hat{\textbf{v}}_{k-1}\nonumber \\ {}&\qquad +\textbf{l}_{k-1}d_{\Gamma }\eta \cdot \nabla d_{\Gamma } -\nu (\theta _0)\mathcal {A}^{k-1}+\int _{-\infty }^\rho \widetilde{\textbf{V}}^{k-1}(r,\cdot )\cdot \nabla d_{\Gamma }\textrm{d}r\bigg )=0, \end{aligned}$$

(3.96)

where $\mathcal {A}^{k-1}$ and $\widetilde{\textbf{V}}^{k-1}$ consist of some terms up to $k-1$ order. Thus

$$\begin{aligned} \hat{p}_{k-1}&=\nu (\theta _0(\rho ))\mathcal {A}^{k-1}-\nu (\theta _0(\rho )){\text {div}}\hat{\textbf{v}}_{k-1} +\big (\textbf{v}_{k-1}^+\eta +\textbf{v}_{k-1}^-(1-\eta )\big )\cdot \nabla d_{\Gamma }\partial _t d_{\Gamma }\nonumber \\ {}&\ -\hat{\textbf{v}}_{k-1}\cdot \nabla d_{\Gamma }\partial _t d_{\Gamma } +2\nu (\theta _0(\rho ))D\hat{\textbf{v}}_{k-1}:\nabla d_{\Gamma }\otimes \nabla d_{\Gamma }+p_{k-1}^+\eta +p_{k-1}^-(1-\eta )\nonumber \\&\ -\left( 2\nu ^+ D\textbf{v}_{k-1}^+\eta + 2\nu ^- D\textbf{v}_{k-1}^-(1-\eta )\right) :\nabla d_{\Gamma }\otimes \nabla d_{\Gamma } \nonumber \\ {}&\ +\eta \int _{+\infty }^\rho {\text {div}}\big (2\nu (\theta _0(r)){\big (D_d\hat{\textbf{v}}\big )_{k-1}}\big )\big )\cdot \nabla d_{\Gamma }\,\textrm{d}r+(1-\eta )\int _{-\infty }^\rho {\text {div}}\big (2\nu (\theta _0(r){\big (D_d\hat{\textbf{v}}\big )_{k-1}}\big )\big )\cdot \nabla d_{\Gamma }\,\textrm{d}r \nonumber \\ {}&\ +\eta \int _\rho ^{+\infty }\widetilde{\textbf{V}}^{k-1}(r,\cdot )\cdot \nabla d_{\Gamma }\, \textrm{d}r-(1-\eta )\int _{-\infty }^\rho \widetilde{\textbf{V}}^{k-1}(r,\cdot )\cdot \nabla d_{\Gamma }\,\textrm{d}r \quad \text {in }\Gamma (3\delta ), \end{aligned}$$

(3.97)

which satisfies the inner–outer matching conditions (3.26). In summary we have:

Lemma 3.10

(The k-th order terms)

Let $k\ge \frac{1}{2}$ and all functions with negative index be supposed to be zero. Then there are smooth functions

$$\begin{aligned} \hat{\textbf{v}}_{k},\textbf{v}_{k}^{\pm },\textbf{u}_{k},\textbf{l}_{k-1},\hat{c}_{k},c_{k}^{\pm },g_{k-\frac{3}{2}},d_{k},\hat{p}_{k-1},p_{k}^{\pm }, \end{aligned}$$

which are bounded on their respective domains, such that for the k-th order the outer equations (3.4), (3.5), (3.6), the inner equations (3.39), (3.40), (3.41), the inner–outer matching conditions (3.26) are satisfied. Moreover, $(\textbf{v}_{k}^{\pm }, p_{k}^{\pm },d_{k})$ satisfies

$$\begin{aligned} \partial _t\textbf{v}_{k}^{\pm }-\nu ^\pm \Delta \textbf{v}_{k}^{\pm }+\nabla p_{k}^{\pm }&=0{} & {} \text {in }\Omega ^{\pm }, \end{aligned}$$

(3.98)

$$\begin{aligned} {\text {div}}\textbf{v}_{k}^{\pm }&=0{} & {} \text {in }\Omega ^{\pm }, \end{aligned}$$

(3.99)

$$\begin{aligned} {\llbracket \textbf{v}_k^\pm \rrbracket \cdot \textbf{n}}&=\hat{a}_{k-\frac{1}{2}}{} & {} \text {on }\Gamma , \end{aligned}$$

(3.100)

$$\begin{aligned} {\llbracket \textbf{v}_k^\pm \rrbracket \cdot {\varvec{\tau }}}&=\check{a}_{k-\frac{1}{2}} -\textbf{u}_0\cdot {\varvec{\tau }}d_{k}{} & {} \text {on }\Gamma , \end{aligned}$$

(3.101)

$$\begin{aligned} \llbracket 2\nu ^\pm D\textbf{v}_{k}^\pm -p_{k}^\pm \textbf{I}\rrbracket \textbf{n}_{\Gamma _t}+\overline{\nu }\llbracket \nabla \textbf{v}_{k}^\pm \rrbracket \textbf{n}_{\Gamma _t}&+\overline{\nu }\llbracket \textbf{v}_{k}^\pm \rrbracket \Delta d_{\Gamma }{} & {} \nonumber \\ +\overline{\nu }\llbracket \textbf{v}_k^\pm \rrbracket \cdot \textbf{n}_{\Gamma _t}\Delta d_{\Gamma }\textbf{n}_{\Gamma _t}&= \sigma \Delta d_{k}\textbf{n}_{\Gamma _t} -\overline{\nu }\llbracket \nabla \textbf{v}_{0}^\pm \rrbracket \nabla d_{k}+\textbf{l}_{0}d_{k}{} & {} \text {on }\Gamma , \end{aligned}$$

(3.102)

$$\begin{aligned} \partial _t d_{k}+\textbf{v}_0^{\pm }\cdot \nabla d_{k}+\tfrac{\textbf{v}_k^{+}+\textbf{v}_k^{-}}{2}\cdot \textbf{n}_{\Gamma _t}-\phi _0d_{k}&=b_{k-\frac{1}{2}}{} & {} \text {on}\ \Gamma , \end{aligned}$$

(3.103)

$$\begin{aligned} \textbf{v}_k^{-}&=\overline{a}_k\textbf{n}_{\partial \Omega },{} & {} \text {on }\partial \Omega , \end{aligned}$$

(3.104)

where $\overline{a}_k = \frac{1}{|\partial \Omega |}\int _{\Gamma _t} (\check{a}_{k-\frac{1}{2}} -\textbf{u}_0\cdot {\varvec{\tau }}d_{k})\,\textrm{d}\sigma$, as well as (3.29). Here (3.101) and (3.102) come from (3.91) and (3.82)(with k instead of $k-1$, $k-\frac{1}{2}$, respectively) and $\hat{a}_{k-\frac{1}{2}}$, $\check{a}_{k-\frac{1}{2}}$, $b_{k-\frac{1}{2}}$ depend only on the terms up to $k-\frac{1}{2}$ order. Furthermore, the compatibility condition (3.91) is satisfied for k instead of $k-1$ and $\textbf{v}_{k}^{\pm }$, $c_{k}^{\pm }$ and $p_{k}^{\pm }$ are extended onto $\Omega ^{\pm }\cup \Gamma (3\delta )$ as in Remark 3.1.

Proof

The lemma is proved by mathematical induction with respect to $k\in \frac{1}{2}\mathbb {N}$, where the beginning of the induction is given by Lemma 3.9. In Theorem A.14 in the appendix we will show solvability of the system (3.97)–(3.103), which will be smooth due to Remark A.2. In the induction hypothesis we assume that

$$\begin{aligned} \{ (\hat{\textbf{v}}_{i},\textbf{v}_{i}^{\pm },\textbf{u}_{i},\textbf{l}_{i-1},\hat{c}_{i},g_{i-\frac{3}{2}},d_{i},\hat{p}_{i-1},p_{i}^{\pm }): 0\le i\le k-\tfrac{1}{2} \} \end{aligned}$$

are known and satisfy the statements of the lemma with i instead of k for all $0\le i\le k-\frac{1}{2}$. Then we obtain the terms for $i=k$ by the following four steps:

Step 1 By the induction hypothesis $\mathcal {S}_{1,k-\frac{3}{2}}$ is known. Since $d_{k-\frac{1}{2}}$ solves (3.102) with $k-\frac{1}{2}$ instead of k, the compatibility condition for (3.41) on $\Gamma$ is satisfied. Moreover, defining $g_{k-\frac{3}{2}}$ by (3.83) the compatibility condition for (3.41) are satisfied on $\Gamma (3\delta )$. Hence we can determine $\hat{c}_k$ as the solution of (3.41) for all $\rho \in \mathbb {R}$, $(x,t)\in \Gamma (3\delta )$.

Step 2 We have seen that the compatibility condition (3.84) for solving (3.39) is equivalent to (3.91) and (3.92). Here (3.91) is satisfied since $(\textbf{v}^\pm _{k-1}, p^\pm _{k-1}, d_{k-1})$ solve (3.97)–(3.103) by assumption in which we have used (3.63) to rewrite (3.91) as (3.101). Moreover, if we define $\textbf{l}_{k-1}$ by (3.92), the compatibility conditions for (3.39) are satisfied. Now we can determine $\hat{\textbf{v}}_{k}$ by (3.39) on $\Gamma$ uniquely. (Note that (3.39) determines $\hat{\textbf{v}}_k$ on $\Gamma (3\delta )\setminus \Gamma$ up to $\textbf{u}_k$, which is not determined yet.) Moreover, this determines $\hat{p}_{k-1}$ by (3.96) on $\Gamma (3\delta )$ since (3.92) holds and $\mathcal {A}^{k-1}$, $\widetilde{\textbf{V}}^{k-1}$ are known.

Step 3 Since $\hat{p}_{k-1}$ is determined, $\mathcal {S}_{1,k-1}$ is known on $\Gamma$ and we can determine $(\textbf{v}^\pm _k, p^\pm _k, d_k)$ as solution of (3.97)–(3.103), cf. Theorem A.14 in the appendix.

Step 4 Using that $(\textbf{v}_k^\pm , p^\pm _k)$ are known, $\textbf{u}_k$ is now determined uniquely by (3.93) and we can determine $\hat{\textbf{v}}_k$ by (3.39) on $\Gamma (3\delta )$ uniquely.

Step 5 Using that $d_k$ is determined on $\Gamma$, one can integrate (3.29) in normal direction to determine $d_k$ uniquely on $\Gamma (3\delta )$.

Finally, we note that $(\hat{\textbf{v}}_k, \hat{p}_{k-1}, \hat{c}_k)$ satisfy the matching conditions on $\Gamma (3\delta )$ by construction, in particular because of the choice of $\textbf{u}_k$. $\square$

3.4 Summary of the construction

The result of this section can be summarized as follows:

Theorem 3.11

Let $N\in \frac{1}{2}\mathbb {N}$. Then there are smooth $(\tilde{c}_A^{in},\tilde{\textbf{v}}_A^{in},\tilde{p}_A^{in})$ defined in $\Gamma (3\delta )$ and smooth $({\tilde{c}_A^\pm }, \tilde{\textbf{v}}_A^\pm , \tilde{p}_A^\pm )$ defined on $\overline{\Omega }\times [0,T_0]$ such that:

• 1. Inner expansion: In $\Gamma (3\delta )$ we have

  $$\begin{aligned} \nonumber \partial _t \tilde{\textbf{v}}_A^{in}+ \tilde{\textbf{v}}_A^{in}\cdot \nabla \tilde{\textbf{v}}_A^{in} -{\text {div}}(2\nu (\tilde{c}_A^{in})D \tilde{\textbf{v}}_A^{in}) +\nabla \tilde{p}_A^{in}&= -\varepsilon {\text {div}}(\nabla \tilde{c}_A^{in}\otimes \nabla \tilde{c}_A^{in}) + R_\varepsilon , \\ \nonumber {\text {div}}\tilde{\textbf{v}}_A^{in}&=\, G_\varepsilon ,\\ \partial _t \tilde{c}_A^{in} + \tilde{\textbf{v}}_A^{in}\cdot \nabla \tilde{c}_A^{in}&= \varepsilon ^{\frac{1}{2}}\Delta \tilde{c}_A^{in} -\varepsilon ^{-\frac{3}{2}} f'(\tilde{c}_A^{in})+ s_\varepsilon , \end{aligned}$$

  (3.105)

  where

  $$\begin{aligned} \Vert (R_\varepsilon , \partial _tG_\varepsilon , s_\varepsilon )\Vert _{L^\infty (\Gamma (3\delta )))}&\le C\varepsilon ^{N+1}, \end{aligned}$$

  (3.106)
  $$\begin{aligned} \Vert G_\varepsilon \Vert _{L^\infty (\Gamma (3\delta ))}&\le C\varepsilon ^{N+2}. \end{aligned}$$

  (3.107)

• 2. Outer expansion: In $\Omega ^{\pm }$ we have ${\tilde{c}_{{A}^{\pm }}} \equiv \pm 1$ and

  $$\begin{aligned} \partial _t {\tilde{\textbf{v}}_A^\pm }+ {\tilde{\textbf{v}}_A^\pm }\cdot \nabla {\tilde{\textbf{v}}_A^\pm } -\nu ^\pm \Delta {\tilde{\textbf{v}}_A^\pm } +\nabla \tilde{p}_A^\pm&= R^\pm _\varepsilon , \\ \nonumber {\text {div}}{\tilde{\textbf{v}}_A^\pm }&=\, 0,\\ \tilde{\textbf{v}}_A^\pm |_{\partial \Omega }&=\, \overline{a}_\varepsilon \textbf{n}_{\partial \Omega }\quad \text {on }\partial \Omega \times [0,T_0], \end{aligned}$$

  (3.108)

  where $\overline{a}_\varepsilon :[0,T]\rightarrow \mathbb {R}$ is smooth and

  $$\begin{aligned} \Vert R_\varepsilon ^\pm \Vert _{L^\infty (\Omega \times [0,T_0])}\le C\varepsilon ^{N+2}\qquad \text {for all }\varepsilon \in (0,1). \end{aligned}$$

• 3. Matching condition: For every $\beta \in \mathbb {N}_0^n$ we have for some $\alpha >0$, $C(M)>0$

  $$\begin{aligned} \begin{aligned}&\Vert \partial _x^{\beta }( \tilde{\textbf{v}}_A^{in}-\tilde{\textbf{v}}_A^{+} \chi _+-\tilde{\textbf{v}}_A^{-}\chi _-)\Vert _{L^\infty (\Gamma (3\delta )\setminus \Gamma (\delta ) )}\le C(M) e^{-\frac{\alpha \delta }{2\varepsilon }},\\&\Vert \partial _x^{\beta }( \tilde{p}_A^{in}-\tilde{p}_A^{+} \chi _+-\tilde{p}_A^{-}\chi _-)\Vert _{L^\infty (\Gamma (3\delta )\setminus \Gamma (\delta ) )}\le C(M) e^{-\frac{\alpha \delta }{2\varepsilon }},\\&\Vert \partial _x^{\beta }(\tilde{c}_A^{in}-{\tilde{c}_A^{+}} \chi _+-{\tilde{c}_A^{-}}\chi _-)\Vert _{L^\infty (\Gamma (3\delta )\setminus \Gamma (\delta ) )}\le C(M) e^{-\frac{\alpha \delta }{2\varepsilon }} \end{aligned} \end{aligned}$$

  for all $\varepsilon \in (0,1)$.

Proof

We define

$$\begin{aligned} {\tilde{c}_A^\pm }(x,t)&=\sum _{k\in \frac{1}{2} \mathbb {N}_0, k\le N+2}\varepsilon ^{k}c_{k}^{\pm }(x,t),\quad \tilde{\textbf{v}}_A^{\pm }(x,t) =\sum _{k\in \frac{1}{2} \mathbb {N}_0, k\le N+2}\varepsilon ^{k}\textbf{v}_{k}^{\pm }(x,t), \\ \tilde{p}_A^{\pm }(x,t)&= \sum _{k\in \frac{1}{2} \mathbb {N}_{-1}, k\le N+2}\varepsilon ^{k}p_{k}^{\pm }(x,t), \end{aligned}$$

and

$$\begin{aligned} \tilde{c}_A^{in}(x,t)&=\sum _{k\in \frac{1}{2} \mathbb {N}_0, k\le N+2}\varepsilon ^{k}\hat{c}_{k}(\tfrac{d_\varepsilon }{\varepsilon },x,t),\quad \tilde{\textbf{v}}_A^{in}(x,t) =\sum _{k\in \frac{1}{2} \mathbb {N}_0, k\le N+2}\varepsilon ^{k}\hat{\textbf{v}}_{k}(\tfrac{d_\varepsilon }{\varepsilon },x,t), \\ \tilde{p}_A^{in}(x,t)&= \sum _{k\in \frac{1}{2} \mathbb {N}_{-1}, k\le N+2}\varepsilon ^{k}\hat{p}_{k}(\tfrac{d_\varepsilon }{\varepsilon },x,t), \end{aligned}$$

where

$$\begin{aligned} d_A(x,t)=\sum _{k\in \frac{1}{2} \mathbb {N}_0, k\le N+2}\varepsilon ^{k}d_k(x,t). \end{aligned}$$

(3.109)

as well as $\overline{a}_\varepsilon (t) = \sum _{k\in \frac{1}{2} \mathbb {N}_0, k\le N+2}\varepsilon ^{k}\overline{a}_{k}(t)$. From the construction one can verify the statements of Theorem 3.11 in the same way as e.g. in [7, Section 4]. $\square$

Remark 3.12

We note that $d_A$ defined in (3.109) satisfies

$$\begin{aligned} |\nabla d_A|^2= 1+ O(\varepsilon ^{N+3})\qquad \text {as }\varepsilon \rightarrow 0 \end{aligned}$$

(3.110)

with respect to $C^k(\Gamma (3\delta ))$ for every $k\in \mathbb {N}$ by the construction (3.29).

4 Refined approximate solutions

In this section we refine the approximate solutions constructed in the previous section by adding a few terms to obtain:

Theorem 4.1

Let $M>0$, $\textbf{u}=\textbf{u}(\varepsilon )\in L^2(0,T_\varepsilon ;H^1(\Omega )^2\cap L^2_\sigma (\Omega ))$ be given for some $T_\varepsilon \in (0,T_0]$, $\varepsilon \in (0,1)$. Moreover, let

$$\begin{aligned} P_M(\textbf{u}) = {\left\{ \begin{array}{ll} \textbf{u} &{}\text {if } \Vert \textbf{u} \Vert _{L^2(0,T_\varepsilon ; H^1(\Omega ))}+ \Vert \textbf{u} \Vert _{H^{\frac{1}{2}}(0,T_\varepsilon ; L^2(\Omega ))} \le M,\\ \frac{M \textbf{u}}{\Vert \textbf{u} \Vert _{L^2(0,T_\varepsilon ; H^1)}+ \Vert \textbf{u} \Vert _{H^{\frac{1}{2}}(0,T_\varepsilon ; L^2)}} &{}\text {else}. \end{array}\right. } \end{aligned}$$

Then there are $c_A\in H^1(0,T_\varepsilon ;L^2(\Omega ))\cap L^2(0,T_\varepsilon ;H^2(\Omega ))$, $p_A\in L^2(0,T_\varepsilon ;H^1(\Omega ))$, and $\textbf{v}_A\in H^1(0,T_\varepsilon ; V(\Omega )')\cap L^2(0,T_\varepsilon ;H^1(\Omega )^2)$, $\textbf{w}_\varepsilon \in H^1(0,T_\varepsilon ;L^2(\Omega ))\cap L^2 (0,T_\varepsilon ;H^1(\Omega ))$ such that

$$\begin{aligned} \partial _t \textbf{v}_A+ \textbf{v}_A\cdot \nabla \textbf{v}_A -{\text {div}}(2\nu (c_A)D \textbf{v}_A) +\nabla p_A&= -\varepsilon {\text {div}}(\nabla c_A\otimes \nabla c_A) + R_\varepsilon , \end{aligned}$$

(4.1)

$$\begin{aligned} {\text {div}}\textbf{v}_A&=\, G_\varepsilon ,\end{aligned}$$

(4.2)

$$\begin{aligned} \partial _t c_A + \big (\textbf{v}_A+\varepsilon ^{N+\frac{1}{4}}((P_M(\textbf{u})+\textbf{w}_\varepsilon )|_{X_\varepsilon (0,S_\varepsilon ,t)}-\textbf{w}_\varepsilon )\big )\cdot \nabla c_A&= \varepsilon ^{\frac{1}{2}}\Delta c_A -\varepsilon ^{-\frac{3}{2}} f'(c_A) +s_\varepsilon ,\end{aligned}$$

(4.3)

$$\begin{aligned} (\textbf{v}_A,c_A)|_{\partial \Omega }&= (0,-1), \end{aligned}$$

(4.4)

where $s_\varepsilon =s_\varepsilon ^1+s_\varepsilon ^2$ with ${\text {supp}}s^2_\varepsilon \subseteq \Gamma (2\delta )$, ${\text {div}}\textbf{w}_\varepsilon =0$ and

$$\begin{aligned} \Vert R_\varepsilon \Vert _{L^2(0,T; (H^1(\Omega ))'))}&\le C(M) \varepsilon ^{N+\frac{1}{4}}(T^{\frac{1}{2}}+\varepsilon ^{\frac{1}{4}}), \end{aligned}$$

(4.5)

$$\begin{aligned} \Vert G_\varepsilon \Vert _{H^1(0,T_\varepsilon ;L^2(\Omega ))}&\le C(M)\varepsilon ^{N+1},\end{aligned}$$

(4.6)

$$\begin{aligned} \Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T_\varepsilon ;H^1(\Omega ))}+\Vert \textbf{w}_\varepsilon \Vert _{L^\infty (0,T_\varepsilon ;L^2(\Omega ))}&\le C(M)\end{aligned}$$

(4.7)

$$\begin{aligned} \Vert s_\varepsilon ^1\Vert _{L^2(\Omega \times (0,T_\varepsilon ))}\le C(M)\varepsilon ^{N+\frac{1}{2}},\quad \Vert s_\varepsilon ^2\Vert _{L^2(0,T_\varepsilon ; ({V_\varepsilon })')}&\le C(M)\varepsilon ^{N+\frac{3}{4}},\end{aligned}$$

(4.8)

$$\begin{aligned} \Vert s_\varepsilon ^2\Vert _{L^2(\Omega \times (0,T_\varepsilon ))}&\le C(M)\varepsilon ^{N-\frac{1}{4}}, \end{aligned}$$

(4.9)

uniformly in $T\in (0,T_\varepsilon ]$ for some $C(M)>0$ independent of $\varepsilon \in (0,1)$. Moreover, $c_A\equiv \pm 1$ in $\Omega ^\pm {\setminus } \Gamma (3\delta )$, $\partial _t c_A$, $\nabla c_A$ are supported in $\overline{\Gamma (3\delta )}$ and ${\text {supp}}s_\varepsilon \subseteq \overline{\Gamma (5\delta /2)}$.

Remark 4.2

Later we will choose $\textbf{u}=\frac{\textbf{w}_1}{\varepsilon ^{N+\frac{1}{4}}}$, cf. also (5.7).

Let $(\tilde{c}_A^{in},\tilde{\textbf{v}}_A^{in},\tilde{p}_A^{in})$, $({\tilde{c}_A^\pm }, \tilde{\textbf{v}}_A^\pm , \tilde{p}_A^\pm )$, and $\overline{a}_\varepsilon (t)$ be as in Theorem 3.11, i.e., the inner and outer pieces of the approximate solution of the Navier–Stokes/Allen–Cahn system constructed in the previous section. Moreover, let $d_A$ be as in (3.109) and define

$$\begin{aligned} d_\varepsilon (x,t)= d_{\Gamma }(x,t)+ \theta (d_\Gamma (x,t)) (d_A(x,t)-d_{\Gamma }(x,t))\quad \text {for all }(x,t)\in \Gamma (3\delta ), \end{aligned}$$

(4.10)

where $\theta \in C^\infty _0(\mathbb {R})$ with $\theta (r)=1$ if $r\in [-\tfrac{5\delta }{2}, \tfrac{5\delta }{2}]$ and ${\text {supp}}\theta \subseteq (-3\delta ,3\delta )$. Moreover, let

$$\begin{aligned} S_\varepsilon (x,t)= S_0(x,t)+ \varepsilon ^{\frac{1}{2}} S_{1/2}(x,t)+ \varepsilon S_1(x,t)+ \varepsilon ^{\frac{3}{2}} S_{3/2}(x,t)\quad \text {for all }(x,t)\in \Gamma (3\delta ),\nonumber \\ \end{aligned}$$

(4.11)

where $S_{1/2}, S_1, S_{3/2}$ are determined such that

$$\begin{aligned} \nabla S_\varepsilon (x,t)\cdot \nabla d_\varepsilon (x,t)= O(\varepsilon ^2) \qquad \text {in }\Gamma (\tfrac{5\delta }{2}) \end{aligned}$$

with respect to any $C^k$-norm, $k\in \mathbb {N}$. Since $\nabla S_0\cdot \nabla d_\Gamma =0$, this leads to the system of first order partial differential equations

$$\begin{aligned} \nabla S_j\cdot \textbf{n}_{\Gamma }&= -\sum _{k=0,\ldots , j-\frac{1}{2}} \nabla S_k\cdot \nabla d_{k-j}\qquad \text {in } \Gamma (\tfrac{5\delta }{2}) \end{aligned}$$

for $j=\frac{1}{2},1,\frac{3}{2}$, which can be solved together with $S_j|_{\Gamma }=0$ by integration in normal direction/the method of characteristics. Moreover, we extend $S_j$ to $S_j:\Gamma (3\delta )\rightarrow \mathbb {T}^1$ such that ${\text {supp}}S_j \subseteq \Gamma (\delta ')$ for some $\delta '\in (\tfrac{5\delta }{2},3\delta )$. Then the assumptions (2.1)–(2.3) are satisfied with $\eta =\frac{1}{2}$. Moreover, let $\rho$ be defined as in (2.12) in the following. Since $d_\varepsilon =d_A$ in $\Gamma (\tfrac{5\delta }{2})$, the definition of $\rho$ coincides with the definition of $\rho$ in Sect. 3, proof of Theorem 3.11, respectively, in $\Gamma (\tfrac{5\delta }{2})$. In the following we will only use the identities from Sect. 3 in the latter domain.

We will now define the refined approximate solution as

$$\begin{aligned} c_{A}(x,t)&=\zeta \circ d_\Gamma c_A^{in}(x,t)+(1-\zeta \circ d_\Gamma )\left( c_A^+ \chi _+ +c_A^-\chi _- \right) ,\\ \textbf{v}_A(x,t)&=\zeta \circ d_\Gamma \textbf{v}_A^{in}(x,t)+(1-\zeta \circ d_\Gamma )\left( \textbf{v}_A^{+}(x,t) \chi _+ +\textbf{v}_A^-(x,t)\chi _-\right) {- \textbf{N}\bar{a}_\varepsilon (t)},\\ p_A(x,t)&=\zeta \circ d_\Gamma p_A^{in}(x,t)+(1-\zeta \circ d_\Gamma ) \left( p_A^{+}(x,t) \chi _+ +p_A^{-}(x,t)\chi _- \right) , \end{aligned}$$

where $\zeta :\mathbb {R}\rightarrow [0,1]$ is smooth such that ${\text {supp}}\zeta \subseteq [-\frac{5\delta }{2},\frac{5\delta }{2}]$ and $\zeta \equiv 1$ on $[-2\delta ,2\delta ]$, $\textbf{N}:\Omega \rightarrow \mathbb {R}^2$ is a smooth vector field such that $\textbf{N}|_{\partial \Omega }= \textbf{n}_{\partial \Omega }$ and ${\text {supp}}\textbf{N}\cap \Gamma (3\delta )=\emptyset$, $c_A^\pm ={\tilde{c}_A^\pm }=\pm 1$ and $\chi _\pm =\chi _{\Omega _t^\pm }(x)$, and we use the following refined ansatz for the inner and outer expansion

$$\begin{aligned} \begin{aligned} c_A^{in} (x,t)&= \tilde{c}_A^{in}(x,t)+\varepsilon ^{N-\frac{3}{4}} \theta _0'({\rho (x,t)}) \bar{h}_{N,\varepsilon }(S_\varepsilon (x,t),t),\\ \bar{h}_{N,\varepsilon }(s,t)&=h_{N-\frac{3}{4},\varepsilon }(s,t)+\sqrt{\varepsilon }h_{N-\frac{1}{4},\varepsilon }(s,t),\\ \textbf{v}_A^{in} (x,t)&=\tilde{\textbf{v}}_A^{in}(x,t) + \varepsilon ^{N+\frac{1}{4}}\left( \textbf{w}_\varepsilon ^+(x,t){\chi _{\Omega ^{\varepsilon ,+}}}+\textbf{w}_\varepsilon ^-(x,t){\chi _{\Omega ^{\varepsilon ,-}}}\right) \\ p_A^{in} (x,t)&=\tilde{p}_A^{in}(x,t)+ \varepsilon ^{N+\frac{1}{4}}\left( q_\varepsilon ^+(x,t){\chi _{\Omega ^{\varepsilon ,+}}}+q_\varepsilon ^-(x,t){\chi _{\Omega ^{\varepsilon ,-}}}\right) ,\\ \textbf{v}_A^{\pm } (x,t)&=\tilde{\textbf{v}}_A^{\pm }(x,t) +\varepsilon ^{N+\frac{1}{4}}\textbf{w}^\pm _\varepsilon (x,t),\quad p_A^{\pm } (x,t)=\tilde{p}_A^{\pm }(x,t)+\varepsilon ^{N+\frac{1}{4}}q^\pm _\varepsilon (x,t). \end{aligned} \end{aligned}$$

(4.12)

We note that we do not use $\rho$-dependent terms in the extra-terms in $\textbf{v}_A^{in}$ and $p_A^{in}$ of order $\varepsilon ^{N+\frac{1}{4}}$. This ansatz differs significantly from the construction of the other terms. It turned out that it not only simplifies the treatment of several remainder terms. It also provides sufficiently good remainder estimates for our analysis, which we could not obtain before. Here it is essential that $(\textbf{w}_\varepsilon ,q_\varepsilon , h_{N-\frac{3}{4},\varepsilon })$ solve the following linearized two-phase flow system

$$\begin{aligned} \partial _t \textbf{w}_\varepsilon ^\pm +\textbf{w}_\varepsilon ^\pm \cdot \nabla \textbf{v}_0^\pm + \textbf{v}_0^\pm \cdot \nabla \textbf{w}_\varepsilon ^\pm -\nu ^\pm \Delta \textbf{w}_\varepsilon ^\pm&+ \nabla q_\varepsilon ^\pm = 0{} & {} \text {in }\Omega _t^{\varepsilon ,\pm }, t\in (0,T_\varepsilon ),\end{aligned}$$

(4.13)

$$\begin{aligned} {\text {div}}\textbf{w}_\varepsilon ^\pm&= 0{} & {} \text {in }\Omega _t^{\varepsilon ,\pm }, t\in (0,T_\varepsilon ),\end{aligned}$$

(4.14)

$$\begin{aligned} \llbracket \textbf{w}_\varepsilon \rrbracket =0,\quad \llbracket \nu D\textbf{w}_\varepsilon -q_\varepsilon \textbf{I}\rrbracket \cdot \textbf{n}_{\Gamma ^\varepsilon _t}&= \sigma \Delta ^{\Gamma _\varepsilon }h_{N-\frac{3}{4},\varepsilon } \textbf{n}_{\Gamma ^\varepsilon _t}{} & {} \text {on }\Gamma _t^\varepsilon , t\in (0,T_\varepsilon ),\end{aligned}$$

(4.15)

$$\begin{aligned} \textbf{w}_\varepsilon ^-|_{\partial \Omega }&= 0{} & {} \text {on }\partial \Omega \times (0,T_\varepsilon ),\end{aligned}$$

(4.16)

$$\begin{aligned} \textbf{w}_\varepsilon ^\pm |_{t=0}&=0{} & {} \text {in } \Omega ^\pm _0 \end{aligned}$$

(4.17)

together with

$$\begin{aligned}&\partial _t h_{N-\frac{3}{4},\varepsilon }+\textbf{n}_{\Gamma ^\varepsilon _t} \cdot \textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)}-(\partial _tS_\varepsilon +\textbf{v}_0\cdot \nabla S_\varepsilon )|_{X_\varepsilon (0,S_\varepsilon ,t)} \partial _sh_{N-\frac{3}{4},\varepsilon }\\\nonumber&\qquad -\sqrt{\varepsilon } |\nabla S_0|^2|_{X_\varepsilon (0,S_\varepsilon ,t)}\partial _s^2h_{N-\frac{3}{4},\varepsilon } +a_\varepsilon (S_\varepsilon ,t) h_{N-\frac{3}{4},\varepsilon } = - (P_M(\textbf{u})\cdot \nabla d_\varepsilon )|_{X_\varepsilon (0,S_\varepsilon ,t)} \end{aligned}$$

(4.18)

on $\mathbb {T}^1\times (0,T_\varepsilon )$ and $h_{N-\frac{3}{4},\varepsilon }|_{t=0}=0$, where $\textbf{w}_\varepsilon ^\pm = \textbf{w}_\varepsilon |_{\Omega ^{\varepsilon ,\pm }}$, $q_\varepsilon ^\pm = q|_{\Omega ^{\varepsilon ,\pm }}$. Here $a_\varepsilon$ is determined in the proof of Theorem 4.3 below. This system can be considered as a linearization of (1.6)–(1.11) if $\sqrt{\varepsilon } H_{\Gamma _t}$ was added to the right-hand side of (1.10) and $\Omega ^\pm _t, \Gamma _t$ was replaced by $\Omega ^{\varepsilon ,\pm }_t, \Gamma _t^\varepsilon$. The function $h_{N-\frac{1}{4},\varepsilon }$ will be determined in the proof of Theorem 4.3 below.

As in Sect. 3 we extend $\textbf{w}^\pm _\varepsilon$ and $q_\varepsilon ^\pm$ to $\Omega \times (0,T_\varepsilon )$ such that ${\text {div}}\textbf{w}_\varepsilon ^\pm =0$ in $\Omega \times (0,T_\varepsilon )$ and $\textbf{w}_\varepsilon ^\pm \in H^1(0,T_\varepsilon ;L^2(\Omega ))\cap L^2 (0,T_\varepsilon ;H^2(\Omega ))$, $q_\varepsilon ^\pm \in L^2(0,T_\varepsilon ;H^1(\Omega ))$ in a bounded manner. Because of Theorem 2.10, we have the uniform bounds

$$\begin{aligned}&\Vert h_{N-\frac{3}{4},\varepsilon }\Vert _{L^\infty (0,T_\varepsilon ;H^1(\mathbb {T}^1))} + \varepsilon ^{\frac{1}{2}} \Vert h_{N-\frac{3}{4},\varepsilon }\Vert _{H^1(0,T_\varepsilon ;H^{\frac{1}{2}})\cap L^2(0,T_\varepsilon ;H^{\frac{5}{2}})}\nonumber \\ {}&\quad +\varepsilon ^{\frac{1}{4}} \Vert \partial _s^2 h_{N-\frac{3}{4},\varepsilon }\Vert _{L^2((0,T_\varepsilon )\times \mathbb {T}^1)}+\Vert \textbf{w}_\varepsilon \Vert _{H^1(0,T_\varepsilon ;V(\Omega )')}+ \Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T_\varepsilon ;H^1(\Omega ))}\nonumber \\&\quad +\varepsilon ^{\frac{1}{2}}\Vert \textbf{w}_\varepsilon \Vert _{H^1(0,T_\varepsilon ;L^2(\Omega ))}+ \varepsilon ^{\frac{1}{2}} \Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T_\varepsilon ;H^2(\Omega ^{\varepsilon ,\pm }_t))} \qquad \le C(M), \end{aligned}$$

(4.19)

where C(M) does not depend on $T_\varepsilon$. The estimate (4.7) follows from the well-known embedding $L^2(0,T_\varepsilon ;V(\Omega ))\cap H^1(0,T_\varepsilon ;V(\Omega )')\hookrightarrow C^0([0,T_\varepsilon ],L^2(\Omega ))$, where the embedding constant is uniform due to (4.17).

For the following we denote $u_A^{in}:= c_A^{in}- \tilde{c}_A^{in}$ and use $\textbf{v}_A^{in}=\tilde{\textbf{v}}_A^{in}+\varepsilon ^{N+\frac{1}{4}}\textbf{w}_\varepsilon$, where $\textbf{w}_\varepsilon = \textbf{w}_\varepsilon ^+\chi _{\Omega ^{\varepsilon ,+}}+\textbf{w}_\varepsilon ^-\chi _{\Omega ^{\varepsilon ,-}}$. Then we obtain in a straight forward manner

$$\begin{aligned}&\partial _t c_A^{in}+ \textbf{v}_A^{in}\cdot \nabla c_A^{in}-\sqrt{\varepsilon }\Delta c_A^{in} + \varepsilon ^{-\frac{3}{2}}f'(c_A^{in})\nonumber \\&= \partial _t u_A^{in}+ \textbf{v}_A^{in}\cdot \nabla u_A^{in}-\sqrt{\varepsilon }\Delta u_A^{in} + \varepsilon ^{-\frac{3}{2}}f''(\tilde{c}_A^{in}) u_A^{in}+\tilde{s}_A^\varepsilon + \varepsilon ^{N+\frac{1}{4}}\textbf{w}_\varepsilon \cdot \nabla \tilde{c}_A^{in}+ O(\varepsilon ^{N+\frac{1}{2}})\nonumber \\&= \partial _t u_A^{in}+ \textbf{v}_A^{in}\cdot \nabla u_A^{in}-\sqrt{\varepsilon }\Delta u_A^{in} + \varepsilon ^{-\frac{3}{2}}f''(\tilde{c}_A^{in}) u_A^{in}+ \varepsilon ^{N+\frac{1}{4}}\textbf{w}_\varepsilon \cdot \nabla \tilde{c}_A^{in} + O(\varepsilon ^{N+\frac{1}{2}}) \end{aligned}$$

(4.20)

in $L^2(0,T_\varepsilon ;L^2(\Gamma _t(2\delta ))):=L^2(\Gamma (2\delta )\cap ((0,T_\varepsilon )\times \mathbb {R}^2))$. Here $\tilde{s}_A^\varepsilon$ is a term that is quadratic in $u_A^{in}$ times $\varepsilon ^{-\frac{3}{2}}$. Hence $\tilde{s}_A^\varepsilon$ is $O(\varepsilon ^{2N-\frac{3}{2}-\frac{3}{2}+\frac{1}{2}})=O(\varepsilon ^{N+\frac{1}{2}})$ in $L^2(0,T_\varepsilon ;L^2(\Gamma _t(2\delta )))$ if $N\ge 3$ due to (4.19).

For the first terms we have:

Theorem 4.3

Let $u_A^{in}=\varepsilon ^{N-\frac{3}{4}} \theta _0'(\rho ) \bar{h}_{N,\varepsilon }(S_\varepsilon (x,t),t)$ and $h_{N-\frac{3}{4},\varepsilon }$ be as before and define

$$\begin{aligned} \mathcal {R}_\varepsilon :=\partial _t u_A^{in}+ \textbf{v}_A^{in}\cdot \nabla u_A^{in}-\sqrt{\varepsilon }\Delta u_A^{in} + \varepsilon ^{-\frac{3}{2}}f''(\tilde{c}_A^{in}) u_A^{in}+\varepsilon ^{N+\frac{1}{4}}(P_M(\textbf{u})+\textbf{w}_\varepsilon )|_{X_\varepsilon (0,S_\varepsilon ,t)}\cdot \nabla \tilde{c}_A^{in}, \end{aligned}$$

where $\textbf{u}=\textbf{u}(\varepsilon )$ is uniformly bounded in $L^2(0,T_\varepsilon ,H^1(\text{\O })^2)\cap H^{\frac{1}{2}}(0,T_\varepsilon ;L^2(\text{\O })^2)$ for small $\varepsilon$, cf. Remark 4.2. Then there is a choice of $h_{N-\frac{1}{4},\varepsilon }\in X_{T_\varepsilon ,0}$ with bounds as in Theorem 2.8 for $\kappa =\sqrt{\varepsilon }$, $r=0$ such that

$$\begin{aligned} \mathcal {R}_\varepsilon =\varepsilon ^{N-\frac{3}{4}} g_\varepsilon (\rho ,S_\varepsilon ,t) +O(\varepsilon ^{N+\frac{1}{2}})\quad \text { in }L^2(0,T_\varepsilon ;L^2(\Gamma _t(2\delta ))), \end{aligned}$$

where $g_\varepsilon$ satisfies the conditions of Lemma 2.15.

Proof

First of all, let us recall where all the appearing terms come from. By (4.12), it holds $c_A^{in}=\tilde{c}_A^{in}+u_A^{in}$ and $\textbf{v}_A^{in}=\tilde{\textbf{v}}_A^{in}+\varepsilon ^{N+\frac{1}{4}}{\textbf{w}_\varepsilon }$, where the terms

$$\begin{aligned} \tilde{c}_A^{in}=\theta _0(\rho )+{\varepsilon ^{3/2}}c_{3/2}(\rho ,x,t)+{\varepsilon ^{2}} {c_{2}}+\cdots \end{aligned}$$

and ${\tilde{\textbf{v}}_A^{in}}=\hat{\textbf{v}}_0(\rho ,x,t)+\varepsilon ^{1/2}\hat{\textbf{v}}_{1/2}+\cdots$ stem from the inner expansion in Sect. 3.2 and are smooth. Here $u_A^{in}=\varepsilon ^{N-\frac{3}{4}} \theta _0'(\rho ) \bar{h}_{N,\varepsilon }(S_\varepsilon (x,t),t)$ is as in (4.12) with the expansion $\bar{h}_{N,\varepsilon }=h_{N-\frac{3}{4},\varepsilon }+\varepsilon ^{\frac{1}{2}}h_{N-\frac{1}{4},\varepsilon }$. Note that $h_{N-\frac{3}{4},\varepsilon }$ will be determined by a coupled equation as in Theorem 2.10 and $h_{N-\frac{1}{4},\varepsilon }$ will be determined by Theorem 2.8 for $\kappa _\varepsilon =\sqrt{\varepsilon }$. Thus these functions will satisfy the uniform estimates in there for $\kappa =\sqrt{\varepsilon }$. Hence we can already assume the latter estimates to hold and disregard some unimportant higher order terms in the following. Finally, recall the properties and expansion form of $d_\varepsilon ,S_\varepsilon$ from above.

We compute all terms with the chain rule and use Taylor for the $f''$-term. This yields

$$\begin{aligned} \mathcal {R}_\varepsilon&=\varepsilon ^{N-\frac{3}{4}}\left( \theta _0''(\rho )\frac{\partial _td_\varepsilon }{\varepsilon } \bar{h}_{N,\varepsilon }+\theta _0'(\rho )(\partial _t\bar{h}_{N,\varepsilon }+\partial _tS_\varepsilon \partial _s\bar{h}_{N,\varepsilon })\right) \\&\quad +\varepsilon ^{N-\frac{3}{4}}\tilde{\textbf{v}}_A^{in}\cdot \left( \theta _0''(\rho )\frac{\nabla d_\varepsilon }{\varepsilon }\bar{h}_{N,\varepsilon }+ \theta _0'(\rho )\nabla S_\varepsilon \partial _s\bar{h}_{N,\varepsilon }\right) \\&\quad -\varepsilon ^{N-\frac{1}{4}}\left( \theta _0'''(\rho )\frac{|\nabla d_\varepsilon |^2}{\varepsilon ^2}\bar{h}_{N,\varepsilon }+2\theta _0''(\rho )\frac{\nabla d_\varepsilon \cdot \nabla S_\varepsilon }{\varepsilon }\partial _s\bar{h}_{N,\varepsilon } +\theta _0''(\rho )\frac{\Delta d_\varepsilon }{\varepsilon } \bar{h}_{N,\varepsilon } + \theta _0'(\rho )\Delta _{\Gamma _\varepsilon }\bar{h}_{N,\varepsilon }\right) \\&\quad + \varepsilon ^{-\frac{3}{2}}\varepsilon ^{N-\frac{3}{4}}\theta _0'(\rho ) \bar{h}_{N,\varepsilon } \left( f''(\theta _0)+f'''(\theta _0)\left[ \varepsilon ^\frac{3}{2} c_\frac{3}{2}+\varepsilon ^2 c_2\right] \right) \\&\quad +\varepsilon ^{N+\frac{1}{4}}(P_M(\textbf{u})+\textbf{w}_\varepsilon )|_{X_\varepsilon (0,S_\varepsilon ,t)}\cdot \frac{\nabla d_\varepsilon }{\varepsilon } \theta _0'(\rho ) +O(\varepsilon ^{N+\frac{1}{2}})\quad \text { in }L^2(\Gamma (2\delta )\cap ((0,T_\varepsilon )\times \mathbb {R}^2)), \end{aligned}$$

where we used that terms of the form $\varepsilon ^N a(\rho )b_\varepsilon (S_\varepsilon ,t)$ with $a\in \mathcal {R}_{0,\alpha }$ and $b_\varepsilon \in L^2(\mathbb {T}^1\times (0,T_\varepsilon ))$ uniformly bounded with respect to small $\varepsilon$ are $O(\varepsilon ^{N+\frac{1}{2}})$ in $L^2(\Gamma (2\delta ))$ with Lemma 2.6. Moreover, we used that $(P_M(\textbf{u})+\textbf{w}_\varepsilon )|_{X_\varepsilon (0,.)}$ is bounded in $L^2((0,T_\varepsilon )\times \mathbb {T}^1)$, the estimates for $h_{N-\frac{3}{4},\varepsilon }$ and $h_{N-\frac{1}{4},\varepsilon }$ as well as $N\ge 2$ to replace $\textbf{v}_A^{in}$ by $\tilde{\textbf{v}}_A^{in}$ up to the error in $L^2(\Gamma (2\delta ))$ above.

Note that the remaining $\textbf{u}$-term is critical and sits at order $O(\varepsilon ^{N-\frac{3}{4}})$. Actually, the point of the theorem is to generate this term. Therefore the prefactor of $\bar{h}_{N,\varepsilon }$ was chosen to be $\varepsilon ^{N-\frac{3}{4}}$ such that the contribution is also in that order. There are several problems one has to overcome. First, there are contributions of $d_\varepsilon$-terms into orders below the critical one. Since $|\nabla d_\varepsilon |^2=1+O(\varepsilon ^2)$, we can cancel the lowest order contribution of this term with the $f''(\theta _0)$-term. Moreover, $\nabla d_\varepsilon \cdot \nabla S_\varepsilon$ is of order $O(\varepsilon ^2)$. By taking a close look at Sect. 3.2, one can infer that $\partial _td_\varepsilon +\tilde{\textbf{v}}_A^{in}\cdot \nabla d_\varepsilon -\sqrt{\varepsilon }\Delta d_\varepsilon$ only gives a contribution of order $O(\varepsilon )$ because there is some cancellation. These terms are of sum structure with $\sqrt{\varepsilon }$-spacing. Furthermore, one has to expand some remaining terms in $\mathcal {R}_\varepsilon$ depending on (x, t) or $(\rho ,x,t)$ into $(\rho ,S_\varepsilon ,t)$. This can be done by transforming the (x, t)-part with $X_\varepsilon$, using Taylor expansion in the first variable and $d_\varepsilon =\varepsilon \rho$. Therefore we use $\hat{\textbf{v}}_0(\rho ,x,t)=\textbf{v}_0^-(x,t)+(\textbf{v}_0^- -\textbf{v}_0^+)(x,t)\eta (\rho )$. Because of $\textbf{v}_0^-=\textbf{v}_0^+$ on $\Gamma$, the second part is improved by the order $\sqrt{\varepsilon }$ due to a Taylor expansion. Finally, the functions in the expansion of $\bar{h}_{N,\varepsilon }$ should be obtained by solving equations of the form mentioned above, of course with the goal to just leave remainders as stated in the theorem. The goal to have a remainder $r_\varepsilon$ suitable for Lemma 2.15 as stated in the theorem leads to the desired equation for $h_{N-\frac{3}{4},\varepsilon }$ in order to resolve the order $\varepsilon ^{N-\frac{3}{4}}$. The remaining terms contribute formally to the order $\varepsilon ^{N-\frac{1}{4}}$. For this order we intend to use $h_{N-\frac{1}{4},\varepsilon }$. However, one has to take care since not all terms in (2.18) for $h_{N-\frac{1}{4},\varepsilon }$ scale as the right hand side with respect to $\kappa =\sqrt{\varepsilon }$ in the $L^2$-norm. More precisely, the first two terms with $\partial _t$ and $\partial _s$ are scaling worse on their own (but only at the amount of $\varepsilon ^{\frac{1}{4}}$), the others are fine. Hence in the application here we need the same prefactor (depending on $\rho$) for those two terms in the equations we require. By having a look at $\mathcal {R}_\varepsilon$, we see that $\theta _0'$ is the desired prefactor. Hence all terms with derivatives of $h_{N-\frac{1}{4},\varepsilon }$ either have the same $\rho$-prefactor $\theta _0'$ or contribute to the order $O(\varepsilon ^N)$ or higher. Thus we obtain an equation of the form as in Theorem 2.8 for $h_{N-\frac{1}{4},\varepsilon }$. More precisely we have

$$\begin{aligned} \partial _t h_{N-\frac{1}{4},\varepsilon }-(\partial _tS_\varepsilon +\textbf{v}_0\cdot \nabla S_\varepsilon )|_{X_\varepsilon (0,S_\varepsilon ,t)} \partial _sh_{N-\frac{1}{4},\varepsilon }&-\sqrt{\varepsilon } \Delta _{\Gamma _\varepsilon }h_{N-\frac{1}{4},\varepsilon }+\tilde{a}_\varepsilon (S_\varepsilon ,t) h_{N-\frac{1}{4},\varepsilon } = \tilde{g}_\varepsilon , \end{aligned}$$

where $\tilde{a}_\varepsilon$ is smooth (uniformly bounded with respect to small $\varepsilon$, derivatives as well) and $\tilde{g}_\varepsilon$ is uniformly bounded in $L^2(\Gamma (2\delta )\cap ((0,T_\varepsilon )\times \mathbb {R}^2))$. By rewriting the $\partial _t h_{N-\frac{1}{4},\varepsilon }$ and $\partial _sh_{N-\frac{1}{4},\varepsilon }$-terms with the above equation, and estimating the $O(\varepsilon ^N)$ remainders with the aid of Lemma 2.6, we finally get remainders as stated in the theorem. Altogether, this yields the claim. $\square$

Proof of Theorem 4.1

First, we estimate $\overline{a}_\varepsilon :(0,T_0)\rightarrow \mathbb {R}$. To this end we define

$$\begin{aligned} \tilde{\textbf{v}}_A(x,t)=\zeta \circ d_\Gamma \tilde{\textbf{v}}_A^{in}(x,t)+(1-\zeta \circ d_\Gamma )\left( \tilde{\textbf{v}}_A^{+}(x,t) \chi _+ +\tilde{\textbf{v}}_A^-(x,t)\chi _-\right) . \end{aligned}$$

Then

$$\begin{aligned} {\text {div}}\tilde{\textbf{v}}_A&= \zeta \circ d_\Gamma G_\varepsilon + O(e^{-\frac{\alpha \delta }{2\varepsilon }}),\quad \partial _t {\text {div}}\tilde{\textbf{v}}_A = \zeta \circ d_\Gamma (\partial _t G_\varepsilon ) + O(e^{-\frac{\alpha \delta }{2\varepsilon }}) \end{aligned}$$

because of the matching conditions and ${\text {div}}\tilde{\textbf{v}}_A^\pm =0$. Since $\overline{a}_\varepsilon = \textbf{n}_{\partial \Omega } \cdot \tilde{\textbf{v}}_A|_{\partial \Omega }$ only depends on t,

$$\begin{aligned} \overline{a}_\varepsilon = \frac{1}{\mathcal {H}^{d-1}(\partial \Omega )}\int _{\Omega } {\text {div}}\tilde{\textbf{v}}_A \, \textrm{d}x =O(\varepsilon ^{N+2})\quad \text {and}\\ \partial _t \overline{a}_\varepsilon = \frac{1}{\mathcal {H}^{d-1}(\partial \Omega )}\int _{\Omega } \partial _t{\text {div}}\tilde{\textbf{v}}_A \, \textrm{d}x =O(\varepsilon ^{N+1}) \end{aligned}$$

in $L^\infty (0,T_0)$ due to (3.105)–(3.106). The rest of the proof is split into three parts.

Part 1: Error in the divergence equation Because of $\llbracket \textbf{w}_\varepsilon \rrbracket =0$ and ${\text {div}}\textbf{w}_\varepsilon ^\pm =0$ in $\Omega ^{\varepsilon ,\pm }$, we have

$$\begin{aligned} \nonumber&{\text {div}}(\textbf{v}_A(x,t) + \textbf{N}\bar{a}_\varepsilon (t)) \nonumber \\&\quad = \zeta (d_\Gamma (x,t)) G_\varepsilon + \zeta '(d_\Gamma (x,t))\nabla d_\Gamma (x,t)\cdot \left( \textbf{v}_A^{in}(x,t) - \textbf{v}^+_A(x,t)\chi _{\Omega ^+(t)}(x) - \textbf{v}^-_A(x,t)\chi _{\Omega ^-(t)}(x)\right) \nonumber \\&\quad = O(\varepsilon ^{N+1}) \end{aligned}$$

(4.21)

in $H^1(0,T_\varepsilon ;L^2(\Omega ))$ because of (3.105)–(3.106) and the matching condition in Theorem 3.11. Together with the previous estimates for $\overline{a}_\varepsilon$ this shows (4.2) for some (different) $G_\varepsilon :\Omega \times (0,T_\varepsilon )\rightarrow \mathbb {R}$, which is given by the sum of the right-hand side of (4.21) and $-{\text {div}}(\textbf{N}\bar{a}_\varepsilon (t))$.

Part 2: Error in the linear momentum equation First of all,

$$\begin{aligned}&\partial _t \textbf{v}_A +\textbf{v}_A\cdot \nabla \textbf{v}_A -{\text {div}}(2\nu (c_A)D\textbf{v}_A)+ \nabla p_A\\&= \zeta (d_\Gamma ) \left( \partial _t \textbf{v}_A^{in} +\textbf{v}_A^{in}\cdot \nabla \textbf{v}_A^{in} -{\text {div}}(2\nu (c_A^{in})D\textbf{v}_A^{in})+ \nabla p_A^{in}\right) \\&\quad + (1-\zeta (d_\Gamma ))\sum _{\pm } \left( \partial _t \textbf{v}_A^{\pm } +\textbf{v}_A^\pm \cdot \nabla \textbf{v}_A^{\pm } -{\text {div}}(2\nu (c_A^\pm )D\textbf{v}_A^{\pm })+ \nabla p_A^{\pm }\right) \chi _\pm +O(\varepsilon ^{N+1}) \end{aligned}$$

in $L^2(\Omega \times (0,T_\varepsilon ))^2$ because of the matching conditions and $\partial _t \overline{a}_\varepsilon (t)\textbf{N}=O(\varepsilon ^{N+1})$ in $L^\infty (0,T_0)$. Since by the construction

$$\begin{aligned} \partial _t \textbf{v}_A^{\pm } +\textbf{v}_A^\pm \cdot \nabla \textbf{v}_A^{\pm } -{\text {div}}(2\nu (c_A^\pm )D\textbf{v}_A^{\pm })+ \nabla p_A^{\pm } = O(\varepsilon ^{N+\frac{5}{4}}) \quad \text {in }L^2(\Omega ^\pm \setminus \Gamma (2\delta )), \end{aligned}$$

we only have to consider the terms from the inner expansion.

Next by the construction of ${\textbf{w}_\varepsilon }$ and (4.15) we have

$$\begin{aligned}&-{\text {div}}(2\nu (\theta _0(\rho ))D\textbf{w}_\varepsilon ) + \nabla q_\varepsilon = -{\text {div}}(2\nu (\theta _0(\rho ))(\textbf{w}^+_\varepsilon \chi _{\Omega ^{\varepsilon ,+}}+\textbf{w}^-_\varepsilon \chi _{\Omega ^{\varepsilon ,-}}))+ \nabla q_\varepsilon \\&\quad = 2{\text {div}}(\nu ^+\textbf{w}^+_\varepsilon \chi _{\Omega ^{\varepsilon ,+}}+\nu ^-\textbf{w}^-_\varepsilon \chi _{\Omega ^{\varepsilon ,-}})+ \nabla q_\varepsilon \\&\qquad + 2{\text {div}}((\nu (\theta _0)-\nu ^+)\chi _{\Omega ^{\varepsilon ,+}} D\textbf{w}_\varepsilon ^+) + 2{\text {div}}((\nu (\theta _0)-\nu ^-)\chi _{\Omega ^{\varepsilon ,-}} D\textbf{w}_\varepsilon ^-) \\&\quad = (-\nu ^+\Delta \textbf{w}_\varepsilon ^++\nabla q_\varepsilon ^+)\chi _{\Omega ^{\varepsilon ,+}}+(-\nu ^-\Delta \textbf{w}_\varepsilon ^-+\nabla q_\varepsilon ^-)\chi _{\Omega ^{\varepsilon ,-}} -\sigma \delta _{\Gamma ^\varepsilon _t}\otimes \Delta _{\Gamma _t^\varepsilon } h_\varepsilon \circ S_\varepsilon \textbf{n}_\varepsilon + O(\varepsilon ^{\frac{1}{4}}) \end{aligned}$$

in $L^2(0,T_\varepsilon ;(H^1(\Gamma _t(3\delta ))^2)')$, where

$$\begin{aligned} \langle \delta _{\Gamma ^\varepsilon _t}\otimes \Delta _{\Gamma _t^\varepsilon } h_\varepsilon \circ S_\varepsilon \textbf{n}_\varepsilon ,\varvec{\varphi }\rangle := \int _{\Gamma _t^\varepsilon } \Delta _{\Gamma _t^\varepsilon } h_\varepsilon (S_\varepsilon (x,t))\textbf{n}_\varepsilon \cdot \varvec{\varphi }(x)\, \textrm{d}\sigma (x)\quad \text {for all }\varvec{\varphi }\in H^1(\Omega )^2. \end{aligned}$$

Here we have used for the second equality that

$$\begin{aligned} \Vert (\nu (\theta _0)-\nu ^\pm ) D\textbf{w}_\varepsilon ^\pm \Vert _{L^2(\Omega ^{\varepsilon ,\pm })}&\le \Vert \nu (\theta _0)-\nu ^\pm \Vert _{L^2(\mathbb {R}_\pm )} \Vert \sup _{r\in [0,2\delta ]}D\textbf{w}_\varepsilon ^\pm (X_\varepsilon (r,.)\Vert _{L^2((0,T)\times \mathbb {T}^1)}\\&\le C\sqrt{\varepsilon } \Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T;H^1(\Omega ^{\varepsilon ,\pm }_t))}^{\frac{1}{2}}\Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T;H^2(\Omega ^{\varepsilon ,\pm }_t))}^{\frac{1}{2}} = O(\varepsilon ^{\frac{1}{4}}) \end{aligned}$$

due to (2.31)–(2.32) as well as $\textbf{n}_\varepsilon = \textbf{n}_{\Gamma _t^\varepsilon }+ O(\varepsilon ^{N+3})$ due to (3.110). Hence we conclude

$$\begin{aligned}&\partial _t \textbf{w}_\varepsilon +\textbf{w}_\varepsilon \cdot \nabla \textbf{v}_0 +\textbf{v}_0 \cdot \nabla \textbf{w}_\varepsilon -{\text {div}}(2\nu (\theta _0(\rho ))D\textbf{w}_\varepsilon )+\nabla q_\varepsilon \\&= -\sigma \delta _{\Gamma ^\varepsilon _t}\otimes \Delta _{\Gamma _t^\varepsilon } h_\varepsilon \circ S_\varepsilon \textbf{n}_\varepsilon \\&\quad +\sum _{\pm } \chi _{\Omega ^{\varepsilon ,\pm }} \left( \partial _t \textbf{w}_\varepsilon ^\pm +\textbf{w}_\varepsilon ^\pm \cdot \nabla \textbf{v}_0^\pm + \textbf{v}_0^\pm \cdot \nabla \textbf{w}_\varepsilon ^\pm -\nu ^\pm \Delta \textbf{w}_\varepsilon ^\pm + \nabla q_\varepsilon ^\pm \right) +O(\varepsilon ^{\frac{1}{4}}) \\&= -\sigma \delta _{\Gamma ^\varepsilon _t}\otimes \Delta _{\Gamma _t^\varepsilon } h_\varepsilon \circ S_\varepsilon \textbf{n}_\varepsilon +O(\varepsilon ^{\frac{1}{4}}) \end{aligned}$$

in $L^2(0,T_\varepsilon ; (H^1(3\delta ))')$. Moreover, we have because of Lemma 2.4 for $\varvec{\varphi }\in H^1(\Gamma _t(3\delta ))^2$

$$\begin{aligned}&\left| \sigma \langle \delta _{\Gamma _t^\varepsilon }\otimes (\Delta _{\Gamma _t^\varepsilon } h_{N-\frac{3}{4},\varepsilon })\circ S_\varepsilon \textbf{n}_\varepsilon , \varvec{\varphi }\rangle - \int _{\Gamma _t^\varepsilon (\frac{11}{4}\delta )} \frac{\theta '_0(\rho )^2}{\varepsilon }(\Delta _{\Gamma _t^\varepsilon } h_{N-\frac{3}{4},\varepsilon })\circ S_\varepsilon \textbf{n}_\varepsilon \cdot \varvec{\varphi }\, \textrm{d}x \right| \\&\quad \le \frac{1}{\varepsilon }\int _{-\frac{11}{4}\delta }^{\frac{11}{4}\delta }\int _{\mathbb {T}^1}| \theta _0'(\tfrac{r}{\varepsilon })^2\textbf{n}_\varepsilon \cdot \left( \varvec{\varphi }(X_\varepsilon (r,s,t))- \varvec{\varphi }(X_\varepsilon (0,s,t)) \right) \Delta _{\Gamma _t^\varepsilon } h_{N-\frac{3}{4},\varepsilon }(s,t)|J_\varepsilon (r,s,t)\, \textrm{d}s\, \textrm{d}r\\&\qquad + C\varepsilon \Vert h_{N-\frac{3}{4},\varepsilon }\Vert _{H^2(\mathbb {T}^1)}\Vert \varvec{\varphi }\Vert _{H^1(\Gamma _t(3\delta ))}\\&\quad \le C\left( \int _{-\frac{11}{4}\delta }^{\frac{11}{4}\delta }|\tfrac{r}{\varepsilon }\theta _0'(\tfrac{r}{\varepsilon })^2|^2\, \textrm{d}r\right) ^{\frac{1}{2}}\Vert \varvec{\varphi }\Vert _{H^1(\Gamma _t(3\delta ))}\Vert \Delta _{\Gamma ^\varepsilon _t} h_{N-\frac{3}{4},\varepsilon }\Vert _{L^2(\mathbb {T}^1)}\\ {}&\qquad + C(M)\varepsilon \Vert h_{N-\frac{3}{4},\varepsilon }\Vert _{H^2(\mathbb {T}^1)}\Vert \varvec{\varphi }\Vert _{H^1(\Gamma _t(3\delta ))}\\&\le C\sqrt{\varepsilon }\Vert \varvec{\varphi }\Vert _{H^1(\Gamma _t(3\delta ))}\Vert h_{N-\frac{3}{4},\varepsilon }\Vert _{H^2(\mathbb {T}^1)} + C(M)\varepsilon \Vert \varvec{\varphi }\Vert _{H^1(\Gamma _t(3\delta ))}\Vert h_{N-\frac{3}{4},\varepsilon }\Vert _{H^2(\mathbb {T}^1)} \end{aligned}$$

since

$$\begin{aligned} \frac{1}{\varepsilon }\int _{-\frac{11}{4}\delta }^{\frac{11}{4}\delta }\theta _0'(\tfrac{r}{\varepsilon })^2J_\varepsilon (r,s,t)\, \textrm{d}r = \frac{\sigma }{|\nabla S_\varepsilon (X_\varepsilon (0,s,t))|} + O(\varepsilon ) \end{aligned}$$

due to (2.5), $|\nabla d_\varepsilon |^2= 1 + O(\varepsilon ^2)$, $\nabla d_\varepsilon \cdot \nabla S_\varepsilon =O(\varepsilon ^2)$, $|\nabla S_\varepsilon (X_\varepsilon (0,s,t)|= |\partial _s X_\varepsilon (0,s,t)|^{-1}+O(\varepsilon ^2)$, and a Taylor expansion around $r=0$. Here $\Vert \partial _s^2 h_{N-\frac{3}{4}}\Vert _{L^2(\mathbb {T}^1\times (0,T_\varepsilon ))}=O(\varepsilon ^{-\frac{1}{4}})$ due to (2.31). Hence we obtain

$$\begin{aligned}&\partial _t \textbf{v}_A + \textbf{v}_A\cdot \nabla \textbf{v}_A -{\text {div}}(2\nu (c_A) D\textbf{v}_A) + \nabla p_A\\&\quad = -\zeta (d_\Gamma )\left( \varepsilon {\text {div}}(\nabla \tilde{c}_A^{in}\otimes \nabla \tilde{c}_A^{in}) +\varepsilon ^{N-\frac{3}{4}}(\theta _0'(\rho ))^2(\Delta _{\Gamma _\varepsilon } h_{N-\frac{3}{4}})\circ S_\varepsilon \right) + O(\varepsilon ^{N+\frac{1}{2}}) \end{aligned}$$

in $L^2(0,T_\varepsilon ; (H^1(\Omega )^2)')$ by using (3.105) and the matching condition in Theorem 3.11.

Now we use that

$$\begin{aligned}&\varepsilon \nabla c_A^{in}\otimes \nabla c_A^{in} -\varepsilon \nabla \tilde{c}_A^{in}\otimes \nabla \tilde{c}_A^{in} \\&\quad = \varepsilon ^{N-\frac{7}{4}}2\theta _0''(\rho )\theta _0'(\rho )\textbf{n}_\varepsilon \otimes \textbf{n}_\varepsilon \overline{h}_{N,\varepsilon }(S_\varepsilon (x,t),t) \\&\qquad + \varepsilon ^{N-\frac{3}{4}}\theta _0'(\rho )^2\left( \nabla ^{\Gamma ^\varepsilon } \overline{h}_{N,\varepsilon }(S_\varepsilon (x,t),t)\otimes \textbf{n}_\varepsilon + \textbf{n}_\varepsilon \otimes \nabla ^{\Gamma ^\varepsilon } \overline{h}_{N,\varepsilon }(S_\varepsilon (x,t),t)\right) \\&\qquad + \varepsilon ^{N-\frac{1}{4}} \textbf{r}_\varepsilon (\rho ,x,t)\cdot \mathfrak {a}_\varepsilon (s,t)\qquad \text {in } \Gamma (\tfrac{5}{2}\delta ) \end{aligned}$$

for some $\textbf{r}_\varepsilon \in (\mathcal {R}_{0,\alpha })^N$, $\mathfrak {a}_\varepsilon \in L^\infty (0,T_\varepsilon ;L^2(\mathbb {T}^1))^N$ and $N\in \mathbb {N}$ with uniformly bounded norms in $\varepsilon \in (0,\varepsilon _0)$. Here one uses that $\sqrt{\varepsilon }h_{N-\frac{1}{4},\varepsilon }, \sqrt{\varepsilon }h_{N-\frac{3}{4},\varepsilon } \in L^\infty (0,T_\varepsilon ; W^1_4(\mathbb {T}^1))$ are bounded (because of (2.32) and (2.21)) and that $\partial _s h_{N-\frac{1}{4},\varepsilon }, h_{N-\frac{3}{4},\varepsilon }$ enter at most quadratically. Hence we obtain

$$\begin{aligned}&-\varepsilon {\text {div}}(\nabla c_A^{in}\otimes \nabla c_A^{in}) +\varepsilon {\text {div}}(\nabla \tilde{c}_A^{in}\otimes \nabla \tilde{c}_A^{in})\\&\quad = -\varepsilon ^{N-\frac{11}{4}} \partial _\rho (2\theta _0''(\rho )\theta _0'(\rho ))(\textbf{n}_\varepsilon \cdot \textbf{n}_\varepsilon ) \textbf{n}_\varepsilon \overline{h}_{N,\varepsilon } -\varepsilon ^{N-\frac{7}{4}} 2\theta _0''(\rho )\theta _0'(\rho )\textbf{n}_\varepsilon \cdot \nabla ^{\Gamma _\varepsilon } \overline{h}_{N,\varepsilon } \\&\qquad -\varepsilon ^{N-\frac{7}{4}} 2\theta _0''(\rho )\theta _0'(\rho ){\text {div}}(\textbf{n}_\varepsilon \otimes \textbf{n}_\varepsilon ) \overline{h}_{N,\varepsilon } -\varepsilon ^{N-\frac{7}{4}} 2\theta _0''(\rho )\theta _0'(\rho )\left( \textbf{n}_\varepsilon \cdot \textbf{n}_\varepsilon \nabla ^{\Gamma _\varepsilon } \overline{h}_{N,\varepsilon }+ \textbf{n}_\varepsilon \cdot \nabla ^\Gamma \overline{h}_{N,\varepsilon }\textbf{n}_\varepsilon \right) \\&\qquad - \varepsilon ^{N-\frac{3}{4}}\theta _0'(\rho )^2{\text {div}}\left( \nabla ^{\Gamma _\varepsilon } \overline{h}_{N,\varepsilon }(S_\varepsilon (x,t),t)\otimes \textbf{n}_\varepsilon + \textbf{n}_\varepsilon \otimes \nabla ^{\Gamma _\varepsilon } \overline{h}_{N,\varepsilon }(S_\varepsilon (x,t),t)\right) \\&\qquad + O(\varepsilon ^{N+\frac{1}{4}}(\sqrt{T}+\varepsilon ^{\frac{1}{4}}))\\&\quad = \nabla \pi _\varepsilon -\varepsilon ^{N-\frac{7}{4}} 2\theta _0''(\rho )\theta _0'(\rho ){\text {div}}(\textbf{n}_\varepsilon )\textbf{n}_\varepsilon \overline{h}_{N,\varepsilon } \\&\qquad - \varepsilon ^{N-\frac{3}{4}}\theta _0'(\rho )^2{\text {div}}\left( \nabla ^{\Gamma _\varepsilon } \overline{h}_{N,\varepsilon }(S_\varepsilon (x,t),t)\otimes \textbf{n}_\varepsilon + \textbf{n}_\varepsilon \otimes \nabla ^{\Gamma _\varepsilon } \overline{h}_{N,\varepsilon }(S_\varepsilon (x,t),t)\right) \\&\qquad + O(\varepsilon ^{N+\frac{1}{4}}(\sqrt{T}+\varepsilon ^{\frac{1}{4}}))\\&\quad = \nabla \pi _\varepsilon - \varepsilon ^{N-\frac{3}{4}}\theta _0'(\rho )^2\Delta ^{\Gamma _\varepsilon } h_{N-\frac{3}{4},\varepsilon }(S_\varepsilon (x,t),t)\textbf{n}_\varepsilon + O(\varepsilon ^{N+\frac{1}{4}}(\sqrt{T}+\varepsilon ^{\frac{1}{4}})) \end{aligned}$$

in $L^2(0,T; (H^1(\Gamma _t(\tfrac{5}{2}\delta ))'))$ for all $T\in (0,T_\varepsilon ]$ since $\textbf{n}_\varepsilon \cdot \textbf{n}_\varepsilon = 1+O(\varepsilon ^2)$, $\textbf{n}_\varepsilon \cdot \nabla \textbf{n}_\varepsilon =O(\varepsilon ^2)$, $\textbf{n}_\varepsilon \cdot \nabla S_\varepsilon = O(\varepsilon ^2)$, and $\overline{h}_{N-\frac{3}{4},\varepsilon }\in L^\infty (0,T_\varepsilon , H^1(\mathbb {T}^1))$ and $\varepsilon ^{\frac{3}{8}}\overline{h}_{N-\frac{3}{4},\varepsilon }\in L^4(0,T_\varepsilon , H^2(\mathbb {T}^1))$ are uniformly bounded, where $\pi _\varepsilon =\varepsilon ^{N-\frac{7}{4}} 2\theta ''_0(\rho )\theta _0 (\rho ) \overline{h}_{N,\varepsilon }$. Now replacing $p_A$ by $p_A+\pi _\varepsilon$ we obtain (4.1).

Part 3: Error in the Allen–Cahn equation Since $c_A^\pm \equiv \pm 1$, the equation (4.3) together with (4.7) and (4.8), follows in a straight forward manner from (4.20), Theorem 4.3 and the matching conditions. $\square$

5 Sharp interface limit

The proof of our main result Theorem 1.1 follows the same steps as in [3, Section 4]. But there are several careful adaptions needed since for our choice of mobility certain estimates “degenerate”/give worse estimates compared to [3] and the construction of the approximate solution is different.

5.1 The leading error in the velocity

For the following let $(c_A, \textbf{v}_A, p_A)$ and $(\tilde{c}_A,\tilde{\textbf{v}}_A, \tilde{p}_A)$ are given as in Sect. 4, where $(c_A, \textbf{v}_A, p_A)$ still depends on the choice of $\textbf{u}$, which will be chosen in the following, but $(\tilde{c}_A,\tilde{\textbf{v}}_A, \tilde{p}_A)$ are independent of $\textbf{u}$. Moreover, we define ${\overline{\textbf{v}}_\varepsilon }:= \textbf{v}_\varepsilon -\textbf{v}_A$. Hence we obtain

$$\begin{aligned} \begin{aligned} \partial _t {\overline{\textbf{v}}_\varepsilon }{+ \textbf{v}_\varepsilon \cdot \nabla {\overline{\textbf{v}}_\varepsilon }}-{\text {div}}(2\nu (c_\varepsilon ) D{\overline{\textbf{v}}_\varepsilon })+\nabla q&={- {\overline{\textbf{v}}_\varepsilon }\cdot \nabla \textbf{v}_A}+{\text {div}}(2(\nu (c_\varepsilon )-\nu (c_A)) D\textbf{v}_A)\\&\quad -\varepsilon {\text {div}}(\nabla {\overline{c}_\varepsilon }\otimes ^s \nabla c_A)-\varepsilon {\text {div}}(\nabla {\overline{c}_\varepsilon }\otimes \nabla {\overline{c}_\varepsilon })- R_\varepsilon ,\\ {\text {div}}{\overline{\textbf{v}}_\varepsilon }&= -G_\varepsilon ,\\ {\overline{\textbf{v}}_\varepsilon }|_{t=0}&=\textbf{v}_{0,\varepsilon } - \textbf{v}_{A}|_{t=0},\\ {\overline{\textbf{v}}_\varepsilon }|_{\partial \Omega }&= 0, \end{aligned} \end{aligned}$$

(5.1)

for some $q:\Omega \times [0,T_0]\rightarrow \mathbb {R}$. Here ${\overline{c}_\varepsilon }=c_\varepsilon -c_A$, $a\otimes ^s b=a\otimes b+b\otimes a$ and $R_\varepsilon$, $G_\varepsilon$ are as in Theorem 4.1.

In the following we consider the estimates

$$\begin{aligned} \sup _{0\le t\le \tau } \Vert { \overline{c}_\varepsilon (t)}\Vert _{L^2(\Omega )} + {\varepsilon ^{\frac{1}{4}}}\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^2(\Omega \times (0,\tau )\setminus \Gamma ^\varepsilon (\frac{3\delta }{2}))}&\le R\varepsilon ^{N+\frac{1}{2}},\end{aligned}$$

(5.2a)

$$\begin{aligned} {\varepsilon ^{\frac{1}{4}}}\Vert \nabla _{{{\varvec{\tau }}_\varepsilon }}{\overline{c}_\varepsilon }\Vert _{L^2(\Omega \times (0,T_\varepsilon )\cap \Gamma ^\varepsilon (\frac{3\delta }{2}))} +\varepsilon \Vert \partial _{\textbf{n}_\varepsilon }{\overline{c}_\varepsilon }\Vert _{L^2(\Omega \times (0,\tau )\cap \Gamma ^\varepsilon (\frac{3\delta }{2}))}&\le R\varepsilon ^{N+\frac{1}{2}},\end{aligned}$$

(5.2b)

$$\begin{aligned} \Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^\infty (0,\tau ;L^2(\Omega ))}+\varepsilon ^{\frac{1}{4}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(\Omega \times (0,\tau ))}&\le R\varepsilon ^{N-\frac{3}{2}},\end{aligned}$$

(5.2c)

$$\begin{aligned} \int _0^\tau \int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}|\nabla {\overline{c}_\varepsilon }|^2 +\varepsilon ^{-2} f''(c_A) {\overline{c}_\varepsilon }^2\, \textrm{d}x\, \textrm{d}t&\le R^2\varepsilon ^{2N+\frac{1}{2}} \end{aligned}$$

(5.2d)

for some $\tau =\tau (\varepsilon ) \in (0,T_0]$, $\varepsilon _0\in (0,1]$, and all $\varepsilon \in (0,\varepsilon _0]$, where $R>0$ is chosen such that

$$\begin{aligned} \Vert c_{0,\varepsilon }-c_A|_{t=0}\Vert _{L^2(\Omega )}^2+ \varepsilon ^4\Vert \nabla (c_{0,\varepsilon }-c_A|_{t=0})\Vert _{L^2(\Omega )}^2+\Vert \textbf{v}_{0,\varepsilon }-\textbf{v}_A|_{t=0}\Vert _{L^2(\Omega )}^2\le \frac{R^2}{4}\varepsilon ^{2N+1}e^{-C_LT_0} \nonumber \\ \end{aligned}$$

(5.3)

for all $\varepsilon \in (0,1]$, where $C_L>0$ is the constant from the spectral estimate in Theorem 2.11. We note that compared to [3, Estimates (4.5)] there is an additional factor $\varepsilon ^{\frac{1}{4}}$ in front of the norms for $\nabla (c_\varepsilon -c_A)$ in (5.2a), for $\nabla _{{\varvec{\tau }}_\varepsilon } (c_\varepsilon -c_A)$ in (5.2b), and for $\Delta (c_\varepsilon -c_A)$ in (5.2c) as well as a loss by $\sqrt{\varepsilon }$ in (5.2d).

As in [3, Section 4], we define

$$\begin{aligned} T_{\varepsilon }:=\sup \{\tau \in [0,T_0]: () \ \text {holds true}\}. \end{aligned}$$

(5.4)

and have $T_{\varepsilon }>0$ because of (5.3).

The main goal of this subsection is to obtain the following bound for the error ${\overline{\textbf{v}}_\varepsilon }$ in the velocity, which again is by a factor $\varepsilon ^{\frac{1}{4}}$ worse than the corresponding result in [3]:

Theorem 5.1

Let $M>0$, $c_A, {\overline{\textbf{v}}_\varepsilon }$ be as in (5.1) and ${\overline{c}_\varepsilon }$ satisfy (5.2) for some $R>0$ and $\tau = T_\varepsilon \in (0,T_0]$ and $N\ge 3$. Then there are some $C(R,M)>0$, $C_0(R)$ independent of $\varepsilon \in (0,\varepsilon _1)$, where $\varepsilon _1$ is as in Theorem 2.1, and $T\in (0,T_\varepsilon ]$ and ${\overline{\textbf{v}}_\varepsilon }=\textbf{w}_1-\textbf{w}_0$, where $\textbf{w}_1\in C([0,T];L^2_\sigma (\Omega ))\cap L^2(0,T;H^1_0(\Omega )^2)\cap H^{\frac{1}{2}}(0,T;L^2_\sigma (\Omega ))$, $\textbf{w}_0\in L^2(0,T;H^2(\Omega )^2)\cap H^1(0,T;L^2(\Omega )^2)$ satisfy

$$\begin{aligned}&\Vert \partial _t \textbf{w}_1\Vert _{L^2(0,T;V(\Omega )')}+\Vert \textbf{w}_1\Vert _{H^{\frac{1}{2}}(0,T;L^2(\Omega ))}+\Vert \textbf{w}_1\Vert _{L^\infty (0,T;L^2(\Omega ))}+\Vert \textbf{w}_1\Vert _{L^2(0,T;H^1(\Omega ))}\nonumber \\&\qquad \le C_0(R)\varepsilon ^{N+\frac{1}{4}}+ C(R,M) \varepsilon ^{N+\frac{1}{4}}(T^{\frac{1}{2}}+ \varepsilon ^{\frac{1}{4}}),\\&\Vert \textbf{w}_0\Vert _{L^2(0,T;H^2(\Omega ))}+ \Vert \partial _t\textbf{w}_0\Vert _{L^2(\Omega \times (0,T)))}\le C(R,M) \varepsilon ^{N+1} \end{aligned}$$

(5.5)

provided that $\Vert \textbf{v}_{0,\varepsilon }-\textbf{v}_A|_{t=0}\Vert _{H^1(\Omega )}\le C \varepsilon ^{N+\frac{1}{2}}$ for some $C>0$.

Remark 5.2

Now we choose

$$\begin{aligned} \textbf{u} = \frac{\textbf{w}_1}{\varepsilon ^{N+\frac{1}{4}}} \in L^2(0,T_0; H^1_0(\Omega )^2)\cap C([0,T_0];L^2_\sigma (\Omega )). \end{aligned}$$

(5.6)

As in [3] this yields a non-linear evolution equation with a globally Lipschitz nonlinearity for $\textbf{u}$, which can be solved in the same manner as in [5, Proof of Lemma 4.2].

The result shows that, if we choose $M=2C_0(R)$, then there are $T'\in (0,T_0]$ and $\varepsilon _0\in (0,1)$ (depending on $R>0$) such that

$$\begin{aligned} \Vert \textbf{u}\Vert _{H^{\frac{1}{2}}(0,T;L^2(\Omega ))}+\Vert \textbf{u}\Vert _{L^\infty (0,T;L^2(\Omega ))}+\Vert \textbf{u}\Vert _{L^2(0,T;H^1(\Omega ))}\le M=2C_0(R) \end{aligned}$$

provided that $T\in (0,\min (T', T_\varepsilon )]$ and $\varepsilon \in (0,\varepsilon _0]$. In particular $P_M(\textbf{u})=\textbf{u}$ in (4.3). After the proof of Theorem 5.1M will be choosen as $M=2C_0(R)$.

Proof of Theorem 5.1

The proof is a variant of the proof of [3, Theorem 4.1]. But there are several careful modifications necessary because of the different powers in the estimates (5.2) in the present case and the new $\varepsilon$-dependent coordinates $(d_\varepsilon , S_\varepsilon )$, which are only approximatively orthogonal.

As in [3] let $(\textbf{w}_0,q_0)$ solve the system

$$\begin{aligned} \partial _t \textbf{w}_0-\Delta \textbf{w}_0+\nabla q_0&=0&\qquad&\text {in }\Omega \times (0,T), \end{aligned}$$

(5.7)

$$\begin{aligned} {\text {div}}\textbf{w}_0&=G_\varepsilon&\qquad&\text {in }\Omega \times (0,T),\end{aligned}$$

(5.8a)

$$\begin{aligned} \textbf{w}_0|_{\partial \Omega }&= 0&\qquad&\text {on }\partial \Omega \times (0,T),\end{aligned}$$

(5.8b)

$$\begin{aligned} \textbf{w}_0|_{t=0}&=(\textbf{v}_A-\textbf{v}_\varepsilon )|_{t=0}&\qquad&\text {in }\Omega , \end{aligned}$$

(5.8c)

where we note that $G_\varepsilon |_{t=0}= {\text {div}}(\textbf{v}_A-\textbf{v}_\varepsilon )|_{t=0}$. By standard results on strong solutions of the Stokes system one obtains a unique solution $\textbf{w}_0\in L^2(0,T;H^2(\Omega )^2)\cap H^1(0,T;L^2(\Omega )^2)$, which satisfies (5.6). Then $\textbf{w}_1:={\overline{\textbf{v}}_\varepsilon }+\textbf{w}_0$ is a solution of the modified system

$$\begin{aligned} \nonumber \partial _t \textbf{w}_1{\!+\! \textbf{v}_\varepsilon \cdot \nabla \textbf{w}_1}\!-\!{\text {div}}(2\nu (c_\varepsilon ) D\textbf{w}_1)\!+\!\nabla q&\!=\!-\!\varepsilon {\text {div}}(\nabla {\overline{c}_\varepsilon }\otimes ^s \nabla c_A)\!-\!\varepsilon {\text {div}}(\nabla {\overline{c}_\varepsilon }\!\otimes \! \nabla {\overline{c}_\varepsilon })\\\nonumber&\quad \!+\!{\text {div}}(2(\nu (c_\varepsilon )\!-\!\nu (c_A)) D\textbf{v}_A) {\!-\!{\overline{\textbf{v}}_\varepsilon }}\cdot \nabla \textbf{v}_A\\&\quad \!+\!\partial _t\textbf{w}_0\!+\!\textbf{v}_\varepsilon \cdot \nabla \textbf{w}_0\!-\!{\text {div}}(2\nu (c_\varepsilon ) D\textbf{w}_0)\!-\! R_\varepsilon ,\\\nonumber {\text {div}}\textbf{w}_1&\!=\! 0,\\\nonumber \textbf{w}_1|_{\partial \Omega }&\!=\! 0,\\\nonumber \textbf{w}_1|_{t=0}&=0 \end{aligned}$$

(5.8d)

in a weak sense. Now testing (5.9) with $\textbf{w}_1$ and using Gronwall’s inequality yields

$$\begin{aligned}&\sup _{0\le t\le T}\frac{1}{2}\int _{\Omega } |\textbf{w}_1|^2(x,t)\textrm{d}x +2\int _{0}^T\int _{\Omega } \nu (c_\varepsilon ) |D\textbf{w}_1|^2\textrm{d}x\,\textrm{d}t\nonumber \\ {}&\quad \le e^{CT}\bigg ( \frac{1}{2}\int _{\Omega } |\textbf{w}_1|^2(x,0)\textrm{d}x+\varepsilon \int _{0}^T\bigg |\int _{\Omega }\big (\nabla {\overline{c}_\varepsilon }\otimes ^s \nabla c_A:\nabla \textbf{w}_1\big )\textrm{d}x\bigg |\textrm{d}t\nonumber \\ {}&\qquad +2\int _{0}^T\bigg |\int _{\Omega }\big (\big (\nu (c_\varepsilon )-\nu (c_A)\big ) D\textbf{v}_A:\nabla \textbf{w}_1\big )\textrm{d}x\bigg |\textrm{d}t \nonumber \\&\qquad +\varepsilon \int _{0}^T\bigg |\int _{\Omega }\big (\nabla {\overline{c}_\varepsilon }\otimes \nabla {\overline{c}_\varepsilon }:\nabla \textbf{w}_1\big )\textrm{d}x\bigg |\textrm{d}t +\int _{0}^T\bigg |\int _{\Omega }R_\varepsilon \cdot \textbf{w}_1\textrm{d}x\bigg |\textrm{d}t \nonumber \\ {}&\qquad +\int _{0}^T\bigg |\int _{\Omega }(\textbf{w}_1\cdot \nabla \textbf{v}_A)\cdot \textbf{w}_1\textrm{d}x\bigg |\textrm{d}t + \int _0^T\bigg | \langle \partial _t \textbf{w}_0, \textbf{w}_1\rangle _{V',V} \bigg |\textrm{d}t \nonumber \\ {}&\qquad +\int _{0}^T\bigg |\int _{\Omega }\big ((\textbf{v}_\varepsilon \cdot \nabla \textbf{w}_0+\textbf{w}_0\cdot \nabla \textbf{v}_A )\cdot \textbf{w}_1+\big (2\nu (c_\varepsilon ) D\textbf{w}_0:\nabla \textbf{w}_1\big )\big )\textrm{d}x\bigg |\textrm{d}t\bigg ). \end{aligned}$$

(5.9)

Now we estimate the different terms on the right-hand side separately.

The most important step is to show

$$\begin{aligned}&\varepsilon \int _0^T\bigg |\int _{\Omega }\nabla {\overline{c}_\varepsilon }\otimes \nabla c_A:\nabla \textbf{w}_1\textrm{d}x\bigg |\textrm{d}t \nonumber \\&\quad \le C(R)\varepsilon ^{N+\frac{1}{4}}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}+ C(R,M)\varepsilon ^{N+\frac{1}{4}}(T^{\frac{1}{2}}+\varepsilon ^{\frac{1}{4}})\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}. \end{aligned}$$

(5.10)

To this end we decompose $\Omega$ into $\Omega \backslash \Gamma _t^\varepsilon (\frac{3\delta }{2})$ and $\Gamma _t^\varepsilon (\frac{3\delta }{2})$ and split the integrals accordingly. Then the proof of (5.11) will consist of three parts.

First of all, we have

$$\begin{aligned}&\varepsilon \int _0^{T}\bigg |\int _{\Omega \backslash \Gamma _t^\varepsilon (\frac{3\delta }{2})}\nabla {\overline{c}_\varepsilon }\otimes \nabla c_A:\nabla \textbf{w}_1\textrm{d}x\bigg | \textrm{d}t \nonumber \\ {}&\qquad \le C(M)\varepsilon \Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^2(\Omega \times (0,T)\backslash \Gamma (\frac{3\delta }{2}))\big )}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}\nonumber \\&\qquad \le C(R,M)\varepsilon ^{N+\frac{5}{4}}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}. \end{aligned}$$

(5.11)

Furthermore, since

$$\begin{aligned} \big |d_\varepsilon \big |\ge \frac{3\delta }{2}\quad \text {for all } x\in \Gamma _t(3\delta )\backslash \Gamma _t(\tfrac{5}{2}\delta ),\ t\in [0,T], \varepsilon \in (0,\varepsilon _0] \end{aligned}$$

(5.12)

due to (2.4), it holds

$$\begin{aligned} \nabla c_A (x,t)&=\nabla (\zeta \circ d_\Gamma )\left( \hat{c}_A^{in}(\rho ,s,t)-c_A^+\chi _+-c_A^-\chi _- \right) +\varepsilon ^{-1}\zeta \circ d_\Gamma \theta _0'(\rho )\textbf{n}_\varepsilon \nonumber \\ {}&\quad +\zeta \circ d_\Gamma \sum _{k=\frac{3}{2}}^{N+2}\varepsilon ^{k} \nabla _{{\varvec{\tau }}_\varepsilon }\hat{c}_k(\rho ,s,t)+\zeta \circ d_\Gamma \sum _{k=\frac{3}{2}}^{N+2}\varepsilon ^{k-1} \partial _{\rho }\hat{c}_k(\rho ,s,t)\textbf{n}_\varepsilon \nonumber \\&\quad +\varepsilon ^{N-\frac{7}{4}}\theta _0''(\rho )\textbf{n}_\varepsilon h_{N,\varepsilon }\circ S_\varepsilon +\varepsilon ^{N-\frac{3}{4}}\theta _0'(\rho )\nabla ( h_{N,\varepsilon }\circ S_\varepsilon )\nonumber \\&= \varepsilon ^{-1}\zeta \circ d_\Gamma \theta _0'(\rho )\textbf{n}_\varepsilon +\varepsilon ^{N-\frac{3}{4}}\theta _0'(\rho )\nabla ( h_{N,\varepsilon }\circ S_\varepsilon )+O(\varepsilon ^\frac{1}{2}) \nonumber \end{aligned}$$

(5.13)

in $L^\infty (\Gamma (3\delta ))$ because of the matching conditions. Therefore we can estimate

$$\begin{aligned}&\varepsilon \int _0^{T}\bigg |\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\nabla {\overline{c}_\varepsilon }\otimes \nabla c_A:\nabla \textbf{w}_1\textrm{d}x\bigg | \textrm{d}t \nonumber \\ {}&\le \underbrace{\int _0^{T}\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\zeta \circ d_\Gamma \theta _0'(\rho )\partial _{\textbf{n}_\varepsilon } {\overline{c}_\varepsilon }\, \textbf{n}_\varepsilon \otimes \textbf{n}_\varepsilon :\nabla \textbf{w}_1\textrm{d}x\,\textrm{d}t}_{=: I}\nonumber \\&\quad + C(M)\varepsilon ^{N-\frac{5}{4}}\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^2\big (0,T;L^4(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))\big )}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))} \nonumber \\ {}&\quad + C(M)\varepsilon ^{\frac{3}{2}}\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^2\big (0,T;L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))\big )}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}\nonumber \\&\le I + C(R,M)\varepsilon ^{N+\frac{1}{2}}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))} \end{aligned}$$

(5.14)

since $\sqrt{\varepsilon } \partial _s h_{N,\varepsilon }\in L^\infty (0,T;L^4(\mathbb {T}^1))$ is bounded due to (2.21) and (2.32). Because of $\partial _{\textbf{n}_\varepsilon } (\theta _0'(\rho ))^2= \frac{1}{\varepsilon }\partial _\rho (\theta _0'(\rho ))^2$ and Lemma 2.7 and using (2.40), we obtain

$$\begin{aligned} |I|&=\left| \,\int _0^{T}\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\zeta \circ d_\Gamma \theta _{0}'(\rho )\partial _{\textbf{n}_\varepsilon }\left( \varepsilon ^{-\frac{1}{2}}Z(S_\varepsilon ,t)\left( \beta _\varepsilon \theta _{0}'(\rho )+\Psi _\varepsilon (\rho ,S_\varepsilon ,t)\right) +\psi ^{\textbf{R}}\right) \right. \nonumber \\&\qquad \left. \textbf{n}_\varepsilon \otimes \textbf{n}_\varepsilon :\nabla \textbf{w}_1\, \textrm{d}x \,\textrm{d}t\right| \nonumber \\&\le \left| \,\int _0^{T}\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\frac{1}{2}\partial _{\textbf{n}_\varepsilon }\left( \theta _{0}'(\rho )^{2}\right) \varepsilon ^{-\frac{1}{2}}Z(S_\varepsilon ,t)\beta _\varepsilon \textbf{n}_\varepsilon \otimes \textbf{n}_\varepsilon :\nabla \textbf{w}_1\textrm{d}x\,\textrm{d}t\right| \nonumber \\&\quad +\,\left| \int _{0}^{T}\int _{\mathbb {T}^{1}}\int _{-\frac{3\delta }{2\varepsilon }}^{\frac{3\delta }{2\varepsilon }}\theta _{0}'(\rho ) \varepsilon ^{-\frac{3}{2}}Z(s,t)\partial _{\rho }\Psi _\varepsilon (\rho ,s,t)\textbf{n}_\varepsilon \otimes \textbf{n}_\varepsilon :\nabla \textbf{w}_1\circ X_\varepsilon \, J_\varepsilon \left( \varepsilon \rho ,s,t\right) \textrm{d}\rho \,\textrm{d}s\,\textrm{d}t\right| \nonumber \\&\quad +C\left\| \psi ^{\textbf{R}}\right\| _{L^{2}\left( 0,T;H^{1}\left( \Gamma ^\varepsilon _t(\frac{3\delta }{2})\right) \right) }\left\| \textbf{w}_1\right\| _{L^{2}\left( 0,T;H^{1}\left( \Gamma ^\varepsilon _t(\frac{3\delta }{2})\right) \right) }\nonumber \\&\quad +C(M)e^{-\frac{\delta }{2\varepsilon }}\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^2\big (\Gamma ^\varepsilon (\frac{3\delta }{2})\big )}\Vert \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}\nonumber \\&\le C\Vert Z\Vert _{L^2(0,T;H^1(\mathbb {T}^{1}))}\Vert \textbf{w}_1\Vert _{L^2(0,T;H^1(\Omega ))}\nonumber \\&\quad +\frac{C}{\varepsilon }\bigg (\sup _{t\in [0,T]}\sup _{s\in \mathbb {T}^1}\int _{I_{\varepsilon }}|\partial _{\rho } \Psi _\varepsilon (\tilde{\rho },s,t)|^2J(\varepsilon \tilde{\rho },s,t)\textrm{d}\tilde{\rho }\bigg )^{\frac{1}{2}}\Vert Z\Vert _{L^2(0,T;H^1(\mathbb {T}^{1}))}\Vert \textbf{w}_1\Vert _{L^2(0,T;H^1(\Omega ))}\nonumber \\&\quad +C\left\| \psi ^{\textbf{R}}\right\| _{L^{2}\left( 0,T;H^{1}\left( \Gamma ^\varepsilon _t(\frac{3\delta }{2})\right) \right) }\left\| \textbf{w}_1\right\| _{L^{2}\left( 0,T;H^{1}\left( \Gamma ^\varepsilon _t(\frac{3\delta }{2})\right) \right) }\nonumber \\&\quad +C(M)e^{-\frac{\delta }{2\varepsilon }}\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^2\big (0,T;L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))\big )}\Vert \textbf{w}_1\Vert _{L^2\big (0,T;L^2(\Omega )\big )}. \end{aligned}$$

(5.15)

Now using (2.43) and (2.44) we derive

$$\begin{aligned} |I|\le C\bigg (\bigg \Vert \frac{\Lambda _\varepsilon }{\varepsilon }\bigg \Vert _{L^1(0,T)}^{\frac{1}{2}}+\Vert {\overline{c}_\varepsilon }\Vert _{L^2\big (0,T;L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))\big )}\bigg )\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))} , \end{aligned}$$

(5.16)

where

$$\begin{aligned} \Lambda _\varepsilon = \int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\left( \varepsilon |\nabla {\overline{c}_\varepsilon }|^2 + \frac{1}{\varepsilon }f''(c_A^\varepsilon ){\overline{c}_\varepsilon }^2\right) \textrm{d}x. \end{aligned}$$

Altogether (5.2), (5.16), and (5.14) yield

$$\begin{aligned}&\varepsilon \int _0^{T}\bigg |\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\nabla {\overline{c}_\varepsilon }\otimes \nabla \tilde{c}_A:\nabla \textbf{w}_1\textrm{d}x\bigg | \textrm{d}t \nonumber \\ {}&\le C(R)\varepsilon ^{N+\frac{1}{4}} \Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}+C(R,M)\varepsilon ^{N+\frac{1}{2}}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}. \end{aligned}$$

(5.17)

This shows (5.11) because of (5.12) and (5.17).

For the following we will use that by construction

$$\begin{aligned} \textbf{v}_A^{in}(x,t)= \tilde{\textbf{v}}_A^{in}(x,t) + \varepsilon ^{N+\frac{1}{4}} {\textbf{w}_\varepsilon (x,t)}. \end{aligned}$$

Using that $\Vert \textbf{w}_\varepsilon \Vert _{L^2(0,T;H^1(\Omega ))}$ is uniformly bounded and $\hat{\textbf{v}}_0(\rho ,x,t)=\textbf{v}_0^+(x,t)\eta (\rho )+\textbf{v}_0^-(x,t)(1-\eta (\rho ))$ together with $\textbf{v}_0^+|_{\Gamma }=\textbf{v}_0^-|_\Gamma$, one can show similarly as in [3, Estimate (4.28)]

$$\begin{aligned} \nabla \textbf{v}_A = \nabla \tilde{\textbf{v}}_A + \varepsilon ^{N+\frac{1}{4}} \nabla \textbf{w}_\varepsilon , \end{aligned}$$

where $\Vert \nabla \tilde{\textbf{v}}_A\Vert _{L^\infty (\Omega \times (0,T))}$, $\Vert \nabla \textbf{w}_\varepsilon \Vert _{L^2(\Omega \times (0,T))}$, $\sqrt{\varepsilon }\Vert \nabla \textbf{w}_\varepsilon \Vert _{L^2(0,T;L^r(\Omega ))}$ are uniformly bounded for every $1\le r<\infty$. Therefore we obtain

$$\begin{aligned} \nonumber&\int _0^T\bigg |\int _{\Omega }\bigg (\big (\nu (c_\varepsilon )-\nu (c_A)\big ) D\textbf{v}_A:\nabla \textbf{w}_1\bigg )\textrm{d}x\bigg |\textrm{d}t\\\nonumber&\quad \le C(M)\left( T^{\frac{1}{2}}\Vert {\overline{c}_\varepsilon }\Vert _{L^\infty (0,T;L^2)}+\varepsilon ^{N+\frac{1}{4}}\Vert {\overline{c}_\varepsilon }\Vert _{L^\infty (0,T;L^4)}\Vert \nabla \textbf{w}_\varepsilon \Vert _{L^2(0,T;L^4)}\right) \Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}\\&\quad \le C(R,M)(T^{\frac{1}{2}}+\varepsilon )\varepsilon ^{N+\frac{1}{2}} \end{aligned}$$

(5.18)

since $|\nu (c_\varepsilon )-\nu (c_A)|\le \Vert \nu '\Vert _{L^\infty (\mathbb {R})}|{\overline{c}_\varepsilon }|$.

Next we use that

$$\begin{aligned} \Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^4(\Omega \times (0,T))}&\le C\bigg (\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^{\infty }(0,T;L^2(\Omega ))}^{\frac{1}{2}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(\Omega \times (0,T))}^{\frac{1}{2}}+T_0^{\frac{1}{4}}\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^{\infty }(0,T;L^2(\Omega ))}\bigg ) \nonumber \\ {}&\le C(R)\big (\varepsilon ^{N-\frac{13}{8}}+\varepsilon ^{N-\frac{3}{2}}\big )\le C(R)\varepsilon ^{N-\frac{13}{8}}, \end{aligned}$$

(5.19)

which yields

$$\begin{aligned} \nonumber&\varepsilon \int _{0}^T\bigg |\int _{\Omega }\bigg (\nabla {\overline{c}_\varepsilon }\otimes \nabla {\overline{c}_\varepsilon }:\nabla \textbf{w}_1\bigg )\textrm{d}x\bigg |\textrm{d}t\le C\varepsilon \Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^4(0,T;L^4(\Omega ))}^2\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))} \nonumber \\ {}&\quad \le C(R)\varepsilon ^{2N-\frac{9}{4}}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}\le C(R)\varepsilon ^{N+\frac{1}{4}}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))} \end{aligned}$$

(5.20)

due to $N\ge \frac{5}{2}$. Because of (4.5),

$$\begin{aligned} \int _{0}^T\bigg |\int _{\Omega }R_\varepsilon \cdot \textbf{w}_1\textrm{d}x\bigg |\textrm{d}t&\le C(M)\varepsilon ^{N+\frac{1}{4}}\Vert \textbf{w}_1\Vert _{L^\infty (0,T;L^2(\Omega ))}\nonumber \\&\quad + C(R,M)\varepsilon ^{N+\frac{1}{4}}(T^{\frac{1}{2}}+\varepsilon ^{\frac{1}{4}})\Vert \textbf{w}_1\Vert _{L^\infty (0,T;L^2(\Omega ))}. \end{aligned}$$

(5.21)

Using (5.6) one obtains as in [3, Proof of Theorem 4.1]

$$\begin{aligned} \nonumber&\int _0^T \left| \langle \partial _t \textbf{w}_0, \textbf{w}_1\rangle _{V',V} \right| \textrm{d}t +\int _{0}^T\bigg |\int _{\Omega }\bigg ( (\textbf{v}_\varepsilon \cdot \nabla \textbf{w}_0-\textbf{w}_0\cdot \nabla \textbf{v}_A)\cdot \textbf{w}_1 +\big (2\nu (c_\varepsilon ) D\textbf{w}_0:\nabla \textbf{w}_1\big )\bigg )\textrm{d}x\bigg |\textrm{d}t\\&\quad \le C(M)\varepsilon ^{N+\frac{1}{2}}\Vert \nabla \textbf{w}_1\Vert _{L^2(\Omega \times (0,T))}+C(M) \varepsilon ^{N+\frac{1}{2}}\Vert \textbf{w}_1\Vert ^{\frac{1}{2}}_{L^{\infty }(0,T;L^2)}\Vert \textbf{w}_1\Vert ^{\frac{1}{2}}_{L^2(0,T;H^1)}, \end{aligned}$$

(5.22)

Combining (5.17), (5.18), (5.20)–(5.22) and utilizing (1.14), Korn’s and Young’s inequality we conclude

$$\begin{aligned} \Vert \textbf{w}_1\Vert _{L^\infty (0,T;L^2(\Omega ))}+\Vert \textbf{w}_1\Vert _{L^2(0,T;H^1(\Omega ))}\le C(R)\varepsilon ^{N+\frac{1}{4}}+C(R,M)\varepsilon ^{N+\frac{1}{4}}(T^{\frac{1}{2}}+\varepsilon ^{\frac{1}{4}}).\nonumber \\ \end{aligned}$$

(5.23)

Furthermore, by testing (5.9) with $\varvec{\varphi }\in L^2(0,T;V(\Omega ))$ and using the similar arguments as above we arrive at

$$\begin{aligned} \Vert \partial _t\textbf{w}_1\Vert _{L^2(0,T;V(\Omega )')} \le C(R)\varepsilon ^{N+\frac{1}{4}}+C(R,M)\varepsilon ^{N+\frac{1}{4}}(T^{\frac{1}{2}}+\varepsilon ^{\frac{1}{4}}), \end{aligned}$$

which by interpolation leads to

$$\begin{aligned} \Vert \textbf{w}_1\Vert _{H^{\frac{1}{2}}(0,T;L^2(\Omega ))}\le C(R)\varepsilon ^{N+\frac{1}{4}}+C(R,M)\varepsilon ^{N+\frac{1}{4}}(T^{\frac{1}{2}}+\varepsilon ^{\frac{1}{4}}). \end{aligned}$$

(5.24)

Finally, (5.6) and (5.23)–(5.24) yield the desired result. $\square$

Since by definition ${\textbf{u}}=\frac{\textbf{w}_1}{\varepsilon ^{N+\frac{1}{4}}}$, we get for $T\in (0,T_\varepsilon )$

$$\begin{aligned} \Vert \partial _t{\textbf{u}}|_{V(\Omega )}\Vert _{L^2(0,T;V(\Omega )')}+\Vert {\textbf{u}}\Vert _{L^\infty (0,T;L^2(\Omega ))}+\Vert {\textbf{u}}\Vert _{L^2(0,T;H^1(\Omega ))}\le C(R). \end{aligned}$$

(5.25)

Proposition 5.3

For $T\in (0,T_\varepsilon )$ there holds

$$\begin{aligned} \int _0^{T}\bigg |\int _{\Omega }\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}\big )\cdot \nabla c_A{\overline{c}_\varepsilon }\, \textrm{d}x\bigg |\textrm{d}t&\le C(R,T)\varepsilon ^{N+\frac{3}{4}}, \end{aligned}$$

(5.26)

where $C(R,T)\rightarrow 0$ as $T\rightarrow 0$.

Proof

Let

$$\begin{aligned} c^{(0)}(x,t)&=\zeta \circ d_\Gamma (x,t) \theta _0(\rho ) +(1-\zeta \circ d_\Gamma (x,t) )\left( c_A^+(x,t)\chi _+(x,t) +c_A^-(x,t)\chi _-(x,t) \right) \end{aligned}$$

(5.27)

be the leading part of $c_A$. Using that

$$\begin{aligned} \bigg (\theta _0(\rho )-\big (\pm 1\big |_{\Omega ^{\pm }(t)}\big )\bigg )\nabla (\zeta \circ d_\Gamma )=O(e^{-\frac{\alpha \delta }{2\varepsilon }}), \end{aligned}$$

(5.28)

due to (5.13), we obtain

$$\begin{aligned}&\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}\big )\cdot \nabla c^{(0)}{\overline{c}_\varepsilon }\textrm{d}x=J+O(e^{-\frac{\alpha \delta }{2\varepsilon }})\Vert {\textbf{u}}\Vert _{H^1(\Omega )}\Vert {\overline{c}_\varepsilon }\Vert _{L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))}, \end{aligned}$$

(5.29)

where

$$\begin{aligned} J&:= \varepsilon ^{-1}\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\zeta \circ d_\Gamma \big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}\big )\cdot \textbf{n}_\varepsilon \theta _0'(\rho ){\overline{c}_\varepsilon }\, \textrm{d}x. \end{aligned}$$

Now we use that

$$\begin{aligned} \partial _r X_\varepsilon (r,s,t)= \textbf{n}_\varepsilon (X_\varepsilon (0,s,t),t)+O(\varepsilon ^2)= \partial _r X_\varepsilon (r',s,t)+O(\varepsilon ^2) \end{aligned}$$

for all $r,r'\in (-\tfrac{3\delta }{2},\tfrac{3\delta }{2})$, $s\in \mathbb {T}^1, t\in [0,T_0]$ because of (2.9). Therefore we obtain

$$\begin{aligned} J&=\int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^r(\textbf{n}_\varepsilon (X_\varepsilon (r,.)) \cdot \left( \partial _r X_\varepsilon (r',.)\cdot (\nabla {\textbf{u}})(X_\varepsilon (r',.))\right) \textrm{d}r'\, \theta _0'(\tfrac{r}{\varepsilon }) {\overline{c}_\varepsilon }(X_\varepsilon (r,.))J_\varepsilon (r,.)\textrm{d}s\,\textrm{d}r\nonumber \\&=\int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^r(\textbf{n}_\varepsilon (X_\varepsilon (r',.)) \cdot \left( \partial _r X_\varepsilon (r',.)\cdot (\nabla {\textbf{u}})(X_\varepsilon (r',.))\right) \textrm{d}r'\, \theta _0'(\tfrac{r}{\varepsilon }) {\overline{c}_\varepsilon }(X_\varepsilon (r,.))J_\varepsilon (r,.)\textrm{d}s\,\textrm{d}r\nonumber \\&\quad + \int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^r \mathbb {A}_\varepsilon (r,r',s,t):(\nabla {\textbf{u}})(X_\varepsilon (r',.))\textrm{d}r'\, \theta _0'(\tfrac{r}{\varepsilon }) {\overline{c}_\varepsilon }(X_\varepsilon (r,.))J_\varepsilon (r,.)\textrm{d}s\,\textrm{d}r, \end{aligned}$$

(5.30)

where $\mathbb {A}_\varepsilon (r,r',s,t)= O(\varepsilon ^2)$. Now using (2.7) we derive

$$\begin{aligned} J&= - \int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^{r}({\text {div}}{\textbf{u}}) (X_\varepsilon (r',.))\textrm{d}r'\, \theta _0'(\tfrac{r}{\varepsilon }) {\overline{c}_\varepsilon }(X_\varepsilon (r,.))J_\varepsilon (r,.)\textrm{d}s\,\textrm{d}r\nonumber \\&\quad - \int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^{r}\nabla S_\varepsilon (X_\varepsilon (r',.))\cdot (\partial _s X_\varepsilon (r',.)\cdot \nabla {\textbf{u}})(X_\varepsilon (r',.))\textrm{d}r'\, \theta _0'(\tfrac{r}{\varepsilon }) {\overline{c}_\varepsilon }(X_\varepsilon (r,.))J_\varepsilon (r,.)\textrm{d}s\,\textrm{d}r\nonumber \\&\quad + \int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^r \mathbb {A}_\varepsilon (r,r',s,t):(\nabla {\textbf{u}})(X_\varepsilon (r',.))\textrm{d}r'\, \theta _0'(\tfrac{r}{\varepsilon }) {\overline{c}_\varepsilon }(X_\varepsilon (r,.))J_\varepsilon (r,.)\textrm{d}s\,\textrm{d}r, \end{aligned}$$

(5.31)

where

$$\begin{aligned}&\left| \int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^{r}\nabla S_\varepsilon (X_\varepsilon (r,.))\cdot (\partial _s X_\varepsilon (r',.)\cdot \nabla {\textbf{u}})(X_\varepsilon (r',.))\textrm{d}r'\, \theta _0'(\tfrac{r}{\varepsilon }) {\overline{c}_\varepsilon }(X_\varepsilon (r,.))J_\varepsilon (r,.)\textrm{d}s\,\textrm{d}r\right| \\&\quad = \left| \int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^{r} {\textbf{u}}(X_\varepsilon (r',.)) \theta _0'(\tfrac{r}{\varepsilon }) \cdot \partial _s\left( \nabla S_\varepsilon (X_\varepsilon (r',.) {\overline{c}_\varepsilon }( X_\varepsilon (r,.))J_\varepsilon (r,.)\right) \textrm{d}r'\,\textrm{d}s\,\textrm{d}r\right| \\&\quad \le C\varepsilon ^{\frac{1}{2}} \Vert {\textbf{u}}(\cdot ,t)\Vert _{L^2(\Omega )}^{\frac{1}{2}}\Vert {\textbf{u}}(\cdot ,t)\Vert _{H^1(\Omega )}^{\frac{1}{2}}\big (\Vert \nabla _{{\varvec{\tau }}_\varepsilon }{\overline{c}_\varepsilon }\Vert _{L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))}+\Vert {\overline{c}_\varepsilon }\Vert _{L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))}\big ) \end{aligned}$$

because of

$$\begin{aligned} \left| \tfrac{1}{\varepsilon }\int _0^{r}{\textbf{u}}(X_\varepsilon (r,s,t){)}\textrm{d}r'\right|&\le \frac{|r|}{\varepsilon }\Vert {\textbf{u}} (X_\varepsilon (\cdot ,s,t))\Vert _{L^\infty (-\frac{3\delta }{2},\frac{3\delta }{2})}\\&\le \frac{|r|}{\varepsilon }\Vert {\textbf{u}} (X_\varepsilon (\cdot ,s,t))\Vert _{L^2 (-\frac{3\delta }{2},\frac{3\delta }{2})}^{\frac{1}{2}}\Vert {\textbf{u}} (X_\varepsilon (\cdot ,s,t))\Vert _{H^1 (-\frac{3\delta }{2},\frac{3\delta }{2})}^{\frac{1}{2}}. \end{aligned}$$

Similarly we have

$$\begin{aligned}&\left| \int _{-\frac{3\delta }{2}}^{\frac{3\delta }{2}}\int _{\mathbb {T}^1} \frac{1}{\varepsilon }\int _0^{r}({\text {div}}{\textbf{u}}) (X_\varepsilon (r',s,t))\textrm{d}r'\, \theta _0'(\tfrac{r}{\varepsilon }) {\overline{c}_\varepsilon }(X_\varepsilon (r,s,t))J_\varepsilon (r,s,t)\textrm{d}s\,\textrm{d}r\right| \\&\quad \le C \Vert {\text {div}}{\textbf{u}}\Vert _{L^2(\Omega )} \Vert {\overline{c}_\varepsilon }\Vert _{L^2(\Gamma ^\varepsilon _t(\frac{3\delta }{2}))} \end{aligned}$$

due to

$$\begin{aligned} \left| \tfrac{1}{\varepsilon }\int _0^{r}({\text {div}}{\textbf{u}})(X_\varepsilon (r,s,t)\textrm{d}r'\right|&\le \frac{|r|^{\frac{1}{2}}}{\varepsilon }\Vert ({\text {div}}{\textbf{u}}) (X_\varepsilon (\cdot ,s,t))\Vert _{L^2 (-\frac{3\delta }{2},\frac{3\delta }{2})}. \end{aligned}$$

Now using $\Vert {\text {div}}{\textbf{u}}\Vert _{L^2(\Omega \times (0,T_\varepsilon ))} = O(\varepsilon ^{\frac{1}{4}})$, we obtain

$$\begin{aligned} \int _{0}^T |J|\textrm{d}t&\le C(R)T^{\frac{1}{4}}\big (\varepsilon ^{\frac{1}{2}}\Vert \nabla _{{\varvec{\tau }}_\varepsilon }{\overline{c}_\varepsilon }\Vert _{L^2(\Omega \times (0,T))}+T^{\frac{1}{2}}\varepsilon ^{\frac{1}{4}}\Vert {\overline{c}_\varepsilon }\Vert _{L^\infty (0,T;L^2(\Omega ))}\big ) \nonumber + CT^{\frac{1}{2}}\varepsilon ^{N+1} \nonumber \\ {}&\le C(R)T^{\frac{1}{4}}(1+T^{\frac{1}{2}})\varepsilon ^{N+\frac{3}{4}}. \end{aligned}$$

(5.32)

Combining (5.32) and (5.29) we obtain

$$\begin{aligned} \int _{0}^T \bigg |\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})}\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}\big )\cdot \nabla c^{(0)} {\overline{c}_\varepsilon }\, \textrm{d}x\bigg |\textrm{d}t\le C(R,T)\varepsilon ^{N+\frac{3}{4}}, \end{aligned}$$

(5.33)

where $C(R,T)\rightarrow 0$ as $T\rightarrow 0$.

Finally, the corresponding estimate $\nabla (c_A- c^{(0)})$ can be done in a straight forward manner since all terms are of higher order in $\varepsilon$ (by at least a factor $\varepsilon ^{\frac{3}{2}}$) compared to $\nabla c^{(0)}$. This finishes the proof. $\square$

Remark 5.4

Analogous to Proposition 5.3 one can show that

$$\begin{aligned} \int _0^{T}\bigg |\int _{\Omega }\big (\textbf{w}_\varepsilon -\textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)}\big )\cdot \nabla c_A{\overline{c}_\varepsilon }\, \textrm{d}x\bigg |\textrm{d}t&\le C(R,T)\varepsilon ^{N+\frac{3}{4}}, \end{aligned}$$

(5.34)

where $C(R,T)\rightarrow 0$ as $T\rightarrow 0$. To this end one uses the same computations and estimates as in the proof of Proposition 5.3 for $\textbf{w}_\varepsilon$ instead of ${\textbf{u}}$ and the estimate (4.7).

5.2 Proof of the main result Theorem 1.1

In order to estimate the error due to linearization of $\varepsilon ^{-\frac{3}{2}}f'(c)$ we need:

Proposition 5.5

Under the assumptions of Sect. 5.1 we have for every $T\in (0,T_\varepsilon )$

$$\begin{aligned} \int _0^{T}\int _{\Omega } |{\overline{c}_\varepsilon }|^3\textrm{d}x\,\textrm{d}t\le C(R)T^{\frac{1}{2}}\varepsilon ^{3N+\frac{7}{8}}. \end{aligned}$$

(5.35)

Proof

The proof is almost identical to [3, Proposition 4.3] with $\Gamma _t(\delta )$ and $\nabla _{\varvec{\tau }}u$ replaced by $\Gamma _t^\varepsilon (\tfrac{3\delta }{2})$, $\nabla _{{\varvec{\tau }}_\varepsilon } {\overline{c}_\varepsilon }$, respectively. In the present case the power of $\varepsilon$ in the estimate for $\Vert \nabla _{{\varvec{\tau }}_\varepsilon } {\overline{c}_\varepsilon }\Vert _{L^2}^{\frac{1}{2}}$ is decreased by $\frac{1}{8}$, which cause the loss of $\frac{1}{8}$ in the power of $\varepsilon$ in the present case compared to [3, Proposition 4.3]. $\square$

In the following the proof is similar to [3, Section 4.2]. But because of the different powers of $\varepsilon$ in the estimates, some terms are critical compared to [3] and we have to choose additionally $T>0$ sufficiently small to finally control all terms.

First of all, by definition $\textbf{v}_\varepsilon = \textbf{v}_A+ \varepsilon ^{N+\frac{1}{4}}{\textbf{u}}-\textbf{w}_0$. Therefore (1.3) and (4.3) imply

$$\begin{aligned} \nonumber \partial _t {\overline{c}_\varepsilon }&+\textbf{v}_\varepsilon \cdot \nabla {\overline{c}_\varepsilon }+ \varepsilon ^{N+\frac{1}{4}}\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}-(\textbf{w}_\varepsilon -\textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)})\big )\cdot \nabla c_A-\textbf{w}_0\cdot \nabla c_A\\&= \varepsilon ^{\frac{1}{2}}\Delta {\overline{c}_\varepsilon }-\varepsilon ^{-\frac{3}{2}}f''(c_A){\overline{c}_\varepsilon } -\varepsilon ^{-\frac{3}{2}}\mathcal {N}(c_A,{\overline{c}_\varepsilon })- s^1_\varepsilon -s^2_\varepsilon , \end{aligned}$$

(5.36)

where $\mathcal {N}(c_A,{\overline{c}_\varepsilon })=f'(c_\varepsilon )-f'(c_A)-f''(c_A){\overline{c}_\varepsilon }.$ Taking the $L^2(\Omega )$-inner product of (5.36) and ${\overline{c}_\varepsilon }$, using integration by parts we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _{\Omega } {\overline{c}_\varepsilon }^2\textrm{d}x+ \varepsilon ^{\frac{1}{2}}\int _{\Omega }\big ( |\nabla {\overline{c}_\varepsilon }|^2+\frac{f''(c_A)}{\varepsilon ^2}{\overline{c}_\varepsilon }^2\big ) \textrm{d}x\nonumber \\ {}&\le \varepsilon ^{N+\frac{1}{4}}\bigg |\int _{\Omega }\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}-(\textbf{w}_\varepsilon -\textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)})\big )\cdot \nabla c_A{\overline{c}_\varepsilon }\, \textrm{d}x\bigg |+\bigg |\int _{\Omega }\textbf{w}_0\cdot \nabla c_A{\overline{c}_\varepsilon }\, \textrm{d}x\bigg |\nonumber \\ {}&\qquad +\varepsilon ^{-\frac{3}{2}}C\int _{\Omega }|{\overline{c}_\varepsilon }|^3\textrm{d}x+\bigg |\int _{\Omega } s_\varepsilon {\overline{c}_\varepsilon }\, \textrm{d}x\bigg | \end{aligned}$$

(5.37)

because of

$$\begin{aligned} \int _{\Omega } \mathcal {N}(c_A,{\overline{c}_\varepsilon }){\overline{c}_\varepsilon }\textrm{d}x\ge -C\int _{\Omega } |{\overline{c}_\varepsilon }|^3\textrm{d}x. \end{aligned}$$

Application of Lemma 2.11 now yields

$$\begin{aligned} {2}&\frac{1}{2}\frac{d}{dt}\int _{\Omega } {\overline{c}_\varepsilon }^2\textrm{d}x-C_L\varepsilon ^{\frac{1}{2}}\int _\Omega {\overline{c}_\varepsilon }^2 \textrm{d}x +\varepsilon ^{\frac{1}{2}} \int _{\Omega \setminus \Gamma ^\varepsilon _t(\frac{3\delta }{2})} |\nabla {\overline{c}_\varepsilon }|^2\textrm{d}x + \varepsilon ^{\frac{1}{2}} \int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})} |\nabla _{{\varvec{\tau }}_\varepsilon } {\overline{c}_\varepsilon }|^2\textrm{d}x\nonumber \\ {}&\quad \le \varepsilon ^{N+\frac{1}{4}}\bigg |\int _{\Omega }\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}-(\textbf{w}_\varepsilon -\textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)})\big )\cdot \nabla c_A{\overline{c}_\varepsilon }\textrm{d}x\bigg |+\bigg |\int _{\Omega }\textbf{w}_0\cdot \nabla c_A{\overline{c}_\varepsilon }\textrm{d}x\bigg |\nonumber \\ {}&\qquad +C\varepsilon ^{-\frac{3}{2}}\int _{\Omega }|{\overline{c}_\varepsilon }|^3\textrm{d}x+\bigg |\int _{\Omega } s_\varepsilon {\overline{c}_\varepsilon }\, \textrm{d}x\bigg |. \end{aligned}$$

(5.38)

Therefore we obtain

$$\begin{aligned}&\sup _{0\le t\le T}\frac{1}{2}\int _{\Omega } {\overline{c}_\varepsilon }(x,t)^2\textrm{d}x + \varepsilon ^{\frac{1}{2}}\int _{0}^T\int _{\Omega \setminus \Gamma ^\varepsilon _t(\frac{3\delta }{2})} |\nabla {\overline{c}_\varepsilon }|^2\textrm{d}x\,\textrm{d}t + \varepsilon ^{\frac{1}{2}}\int _{0}^T\int _{\Gamma ^\varepsilon _t(\frac{3\delta }{2})} |\nabla _{{\varvec{\tau }}_\varepsilon } {\overline{c}_\varepsilon }|^2\,\textrm{d}x\,\textrm{d}t\nonumber \\&\quad \le e^{C_LT_0}\bigg (\frac{1}{2}\int _{\Omega } {\overline{c}_\varepsilon }(x,0)^2\textrm{d}x +\varepsilon ^{N+\frac{1}{4}}\int _{0}^T\bigg |\int _{\Omega }\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}-(\textbf{w}_\varepsilon -\textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)})\big )\nonumber \\&\qquad \cdot \nabla c_A{\overline{c}_\varepsilon }\,\textrm{d}x\bigg |\textrm{d}t\nonumber \\&\qquad +{\int _0^T}\bigg |\int _{\Omega }\textbf{w}_0\cdot \nabla c_A{\overline{c}_\varepsilon }\,\textrm{d}x\bigg |{\textrm{d}t}+C\varepsilon ^{-\frac{3}{2}}\int _{0}^T\int _{\Omega }|{\overline{c}_\varepsilon }|^3\textrm{d}x\,\textrm{d}t+\int _{0}^T\bigg |\int _{\Omega } s_\varepsilon {\overline{c}_\varepsilon }\, \textrm{d}x\bigg |\textrm{d}t\bigg ) \nonumber \\ {}&\quad \le \frac{R^2}{8}\varepsilon ^{2N+1}+C'(R,T)\big (\varepsilon ^{2N+1}+\varepsilon ^{3N-\frac{5}{8}}\big )\le \frac{R^2}{8}\varepsilon ^{2N+1}+C(R,T)\varepsilon ^{2N+1} \end{aligned}$$

(5.39)

for all $0\le T\le T_\varepsilon$ due to (4.8), (5.2), (5.3), Theorem 5.1, Proposition 5.3, Remark 5.4, Proposition 5.5, $\Vert \nabla c_A\Vert _{L^\infty (0,T_0;L^2(\Omega ))}=O(\tfrac{1}{\sqrt{\varepsilon }})$, Gronwall’s inequality, and $N\ge 3$, where $C(R,T),C'(R,T)\rightarrow _{T\rightarrow 0} 0$. Hence, if $\varepsilon \in \min (0,\varepsilon _0)$ and $\varepsilon _0>0$ and $T>0$ are sufficiently small, we have

$$\begin{aligned} \frac{R^2}{8}\varepsilon ^{2N+1}+C(R,T)\varepsilon ^{2N+1}\le \frac{R^2}{4}\varepsilon ^{2N+1} \end{aligned}$$

and therefore

$$\begin{aligned} \nonumber&\sup _{0\le t\le T}\frac{1}{2}\int _{\Omega } {\overline{c}_\varepsilon }(x,t)^2\textrm{d}x +\varepsilon ^{\frac{1}{2}} \int _{\Omega \times (0,T)\setminus \Gamma ^\varepsilon (\frac{3\delta }{2})} |\nabla {\overline{c}_\varepsilon }|^2\textrm{d}(x,t) \\&\quad + \varepsilon ^{\frac{1}{2}} \int _{\Omega \times (0,T)\cap \Gamma ^\varepsilon (\frac{3\delta }{2})} |\nabla _{{\varvec{\tau }}_\varepsilon } {\overline{c}_\varepsilon }|^2\textrm{d}(x,t)\le \frac{R^2}{4}\varepsilon ^{2N+1}. \end{aligned}$$

(5.40)

Combining this estimate with (5.37) we obtain

$$\begin{aligned}&\varepsilon ^{\frac{1}{2}}\int _0^T \int _{\Omega }\left( |\nabla {\overline{c}_\varepsilon }|^2+\frac{f''(c_A)}{\varepsilon ^2}{\overline{c}_\varepsilon }^2\right) \textrm{d}x\,\textrm{d}t\le \frac{1}{2}\int _{\Omega } {\overline{c}_\varepsilon }(x,0)^2\textrm{d}x \nonumber \\ {}&\quad +\varepsilon ^{N+\frac{1}{4}}\int _{0}^T\bigg |\int _{\Omega }\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}-(\textbf{w}_\varepsilon -\textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)})\big )\cdot \nabla c_A{\overline{c}_\varepsilon }\textrm{d}x\bigg |\textrm{d}t\nonumber \\ {}&\quad +{\int _0^T}\bigg |\int _{\Omega }\textbf{w}_0\cdot \nabla c_A{\overline{c}_\varepsilon }\,\textrm{d}x\bigg |{\textrm{d}t}+C\varepsilon ^{-\frac{3}{2}}\int _{0}^T\int _{\Omega }|{\overline{c}_\varepsilon }|^3\textrm{d}x\,\textrm{d}t+\int _{0}^T\bigg |\int _{\Omega }s_\varepsilon {\overline{c}_\varepsilon }\textrm{d}x\bigg |\textrm{d}t \nonumber \\&\quad \le \frac{R^2}{8}\varepsilon ^{2N+1}+C(R,T)\varepsilon ^{2N+1}\le \frac{R^2}{4}\varepsilon ^{2N+1}. \end{aligned}$$

(5.41)

for $T\in (0, T_\varepsilon ]$ sufficiently small and, if $\varepsilon _0>0$ is sufficiently small,

$$\begin{aligned} \varepsilon ^{2}\int _0^T \int _{\Omega }|\partial _{\textbf{n}_\varepsilon } {\overline{c}_\varepsilon }|^2\textrm{d}x\,\textrm{d}t&\le \varepsilon ^{2}\int _0^T \int _{\Omega }|\nabla {\overline{c}_\varepsilon }|^2\textrm{d}x\, \textrm{d}t \nonumber \\ {}&\le \int _0^T \int _{\Omega }\big ( \varepsilon ^{2}|\nabla {\overline{c}_\varepsilon }|^2+f''(c_A){\overline{c}_\varepsilon }^2\big ) \textrm{d}x\,\textrm{d}t+C\int _0^T \int _{\Omega }{\overline{c}_\varepsilon }^2\textrm{d}x\,\textrm{d}t\nonumber \\ {}&\le \frac{R^2}{4}\varepsilon ^{2N+\frac{5}{2}}+CT\frac{R^2}{2}\varepsilon ^{2N+1} \le \frac{3R^2}{4}\varepsilon ^{2N+1}. \end{aligned}$$

(5.42)

Next we derive the estimates for $\Delta {\overline{c}_\varepsilon }$ in $L^2(\Omega \times (0,T))$ and $\nabla {\overline{c}_\varepsilon }$ in $L^\infty (0,T;L^2(\Omega ))$. To this end we take the $L^2(\Omega )$-inner product of (5.36) and $-\varepsilon ^4\Delta {\overline{c}_\varepsilon }$, integrate by parts, and obtain

$$\begin{aligned}&\sup _{0\le t\le T}\frac{\varepsilon ^4}{2}\int _{\Omega } |\nabla {\overline{c}_\varepsilon }|^2(x,t)\textrm{d}x+ \varepsilon ^{\frac{9}{2}}\int _0^T\int _{\Omega }|\Delta {\overline{c}_\varepsilon }|^2\textrm{d}x\,\textrm{d}t \nonumber \\ {}&\quad \le \frac{\varepsilon ^4}{2}\int _{\Omega } |\nabla {\overline{c}_\varepsilon }|^2(x,0)\textrm{d}x+\varepsilon ^{\frac{5}{2}}\int _0^T\int _{\Omega }\big |f''(c_A){\overline{c}_\varepsilon }\Delta {\overline{c}_\varepsilon }\big |\textrm{d}x\,\textrm{d}t +\varepsilon ^4\int _0^T\int _{\Omega }\big |\textbf{v}_\varepsilon \cdot \nabla {\overline{c}_\varepsilon }\Delta {\overline{c}_\varepsilon } \big |\textrm{d}x\,\textrm{d}t \nonumber \\ {}&\qquad +\varepsilon ^{N+\frac{17}{4}}\int _0^T\int _{\Omega }\big |\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}-(\textbf{w}_\varepsilon -\textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)})\big )\cdot \nabla c_A\Delta {\overline{c}_\varepsilon }\big |\textrm{d}x\,\textrm{d}t \nonumber \\ {}&\qquad +\varepsilon ^4{\int _0^T}\bigg |\int _{\Omega }\textbf{w}_0\cdot \nabla c_A\Delta {\overline{c}_\varepsilon }\,\textrm{d}x\bigg |{\textrm{d}t}+\varepsilon ^{\frac{5}{2}}\int _0^T\int _{\Omega }\big |\mathcal {N}(c_A,{\overline{c}_\varepsilon })\Delta {\overline{c}_\varepsilon }\big |\textrm{d}x\,\textrm{d}t+\varepsilon ^4\int _0^T\bigg |\int _{\Omega }s_\varepsilon \Delta {\overline{c}_\varepsilon }\,\textrm{d}x\bigg |\textrm{d}t, \end{aligned}$$

(5.43)

where

$$\begin{aligned} \varepsilon ^{\frac{9}{2}}\int _0^T\int _{\Omega }\big |f''(c_A){\overline{c}_\varepsilon }\Delta {\overline{c}_\varepsilon }\big |\textrm{d}x\,\textrm{d}t&\le C\varepsilon ^{\frac{9}{2}} T^{\frac{1}{2}}\Vert {\overline{c}_\varepsilon }\Vert _{L^\infty (0,T;L^2(\Omega ))}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))} \nonumber \\ {}&\le CR\varepsilon ^{N+3} T^{\frac{1}{2}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(\Omega \times (0,T))}. \end{aligned}$$

(5.44)

As in [3, Section 4.2] one estimates

$$\begin{aligned}&\varepsilon ^{2}\int _0^T\int _{\Omega }\big |\mathcal {N}(c_A,{\overline{c}_\varepsilon })\Delta {\overline{c}_\varepsilon }\big |\textrm{d}x\,\textrm{d}t \le CR^2\varepsilon ^{2N+2}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))} \end{aligned}$$

(5.45)

and, using $\textbf{v}_\varepsilon = \textbf{v}_A+ \varepsilon ^{N+\frac{1}{4}}{\textbf{u}}-\textbf{w}_0$,

$$\begin{aligned} \varepsilon ^4\int _0^T\int _{\Omega }\big |\textbf{v}_\varepsilon \cdot \nabla {\overline{c}_\varepsilon }\Delta {\overline{c}_\varepsilon } \big |\textrm{d}x\,\textrm{d}t&\le C R\varepsilon ^{N+\frac{7}{2}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))}\nonumber \\ {}&\quad +C(R)\varepsilon ^{N+\frac{17}{4}}\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^\infty (0,T;L^2(\Omega ))}^{\frac{1}{2}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))}^{\frac{3}{2}}. \end{aligned}$$

(5.46)

Furthermore

$$\begin{aligned} \nonumber&\varepsilon ^{N+\frac{9}{2}}\int _0^T\int _{\Omega }\big |\big ({\textbf{u}}-{\textbf{u}}|_{X_\varepsilon (0,S_\varepsilon ,t)}-(\textbf{w}_\varepsilon -\textbf{w}_\varepsilon |_{X_\varepsilon (0,S_\varepsilon ,t)})\big )\cdot \nabla c_A\Delta {\overline{c}_\varepsilon }\big |\textrm{d}x\,\textrm{d}t\\&\qquad \le C(R)\varepsilon ^{N+\frac{13}{4}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))}, \\ {}&\varepsilon ^4{\int _0^T}\bigg |\int _{\Omega }\textbf{w}_0\cdot \nabla c_A\Delta {\overline{c}_\varepsilon }\,\textrm{d}x\bigg |{\textrm{d}t}\le C(R)\varepsilon ^{N+\frac{9}{2}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))} \end{aligned}$$

(5.47)

and

$$\begin{aligned} \nonumber \varepsilon ^4\int _0^T\bigg |\int _{\Omega }s_\varepsilon \Delta {\overline{c}_\varepsilon }\,\textrm{d}x\bigg |\textrm{d}t&\le \varepsilon ^4\Vert s^1_\varepsilon +s^2_\varepsilon \Vert _{L^2(0,T;L^2(\Omega ))}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))}\\&\le C\varepsilon ^{N+\frac{15}{4}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))}. \end{aligned}$$

(5.48)

due to (4.9).

Now using (5.3), (5.44) and (5.46)–(5.49) in (5.43) leads to

$$\begin{aligned}&\sup _{0\le t\le T}\frac{\varepsilon ^4}{2}\int _{\Omega } |\nabla {\overline{c}_\varepsilon }|^2(x,t)\textrm{d}x+ \varepsilon ^{\frac{9}{2}}\int _0^T\int _{\Omega }|\Delta {\overline{c}_\varepsilon }|^2\textrm{d}x\,\textrm{d}t \nonumber \\ {}&\le \frac{R^2}{8}\varepsilon ^{2N+1}+C(R)\varepsilon ^{N+\frac{17}{4}}\Vert \nabla {\overline{c}_\varepsilon }\Vert _{L^\infty (0,T;L^2(\Omega ))}^{\frac{1}{2}}\Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))}^{\frac{3}{2}} \nonumber \\\nonumber&\quad +C(R)\varepsilon ^{N+3} \Vert \Delta {\overline{c}_\varepsilon }\Vert _{L^2(0,T;L^2(\Omega ))}. \end{aligned}$$

(5.49)

An application of Young’s inequality yields

$$\begin{aligned}&\sup _{0\le t\le T}\frac{3\varepsilon ^4}{8}\int _{\Omega } |\nabla {\overline{c}_\varepsilon }|^2(x,t)\textrm{d}x+ \frac{3}{8}\varepsilon ^{\frac{9}{2}}\int _0^T\int _{\Omega }|\Delta {\overline{c}_\varepsilon }|^2\textrm{d}x\,\textrm{d}t \nonumber \\ {}&\qquad \le \frac{R^2}{8}\varepsilon ^{2N+1}+C(R)\varepsilon ^{2N+\frac{3}{2}}\le \frac{R^2}{4}\varepsilon ^{2N+1}, \end{aligned}$$

(5.50)

and

$$\begin{aligned} \sup _{0\le t\le T}\int _{\Omega } |\nabla {\overline{c}_\varepsilon }|^2(x,t)\textrm{d}x + \varepsilon ^{\frac{1}{2}}\int _{0}^T\int _{\Omega } |\Delta {\overline{c}_\varepsilon }|^2\textrm{d}x\, \textrm{d}t \le \frac{2R^2}{3}\varepsilon ^{2N-3}. \end{aligned}$$

(5.51)

if $\varepsilon _0>0$ is small enough.

Altogether we see from (5.40), (5.41), (5.42) and (5.51) and the definition of $T_\varepsilon$ that there are $\varepsilon _0>0$ and $T_1>0$ such that $T_\varepsilon >T_1$ for all $\varepsilon \in (0,\varepsilon _0)$ and therefore (5.2) hold true for $\tau =T_1$.

Finally, (1.18) follows from ${\overline{\textbf{v}}_\varepsilon }=\textbf{v}_\varepsilon -\textbf{v}_A$ and Theorem 5.1, in particular (5.5), and the remaining two conclusions in Theorem 1.1 are a consequence of the constructions of $c_A$ and $\textbf{v}_A$. This finishes the proof of Theorem 1.1.

Wellposedness of linearized two-phase flow system

Theorem A.14

Let $\Omega ^\pm , \Gamma$ be smooth and as in Sect. 1 with $V_{\Gamma _t}= \textbf{n}_{\Gamma _t}\cdot \textbf{v}_0^\pm$ for all $t\in [0,T_0]$ and some smooth $\textbf{v}_0^\pm :\overline{\Omega ^\pm }\rightarrow \mathbb {R}^2$ with ${\text {div}}\textbf{v}_0^\pm =0$ in $\Omega ^\pm$. Moreover, let $T\in (0,T_0]$, $q\in (4,\infty )$, and $\textbf{b}_j:\Gamma \rightarrow \mathbb {R}^2$, $j=0,1,2$, $a_k:\mathbb {T}^1\times [0,T_0]\rightarrow \mathbb {R}$, $k=0,1$ be smooth. Then for all

$$\begin{aligned} \textbf{f}&\in L^q(\Omega \times (0,T))^2, \quad g\in L^q(0,T;W^1_q(\Omega \setminus \Gamma _t)), \quad w \in L^q(0,T;W_q^{2-\frac{1}{q}}(\mathbb {T}^1)), \\ \textbf{a}_1&\in L^q(0,T;W_q^{2-\frac{1}{q}}(\Gamma _t))^2\cap W_q^{\frac{1}{2}-\frac{1}{2q}}(0,T;L^q(\Gamma _t))^2,\quad \textbf{v}_0 \in W^{2-\frac{2}{q}}_q(\Omega \setminus \Gamma _0)^2, \\ \textbf{a}_2&\in L^q(0,T;W_q^{1-\frac{1}{q}}(\Gamma _t))^2\cap W_q^{\frac{1}{2}-\frac{1}{2q}}(0,T;L^q(\Gamma _t))^2, \quad h_0 \in W^{3-\frac{2}{q}}_q(\mathbb {T}^1), \\ \textbf{a}&\in L^q(0,T;W^{2-\frac{1}{q}}_q(\partial \Omega ))^2\cap W_q^{1-\frac{1}{2q}}(0,T;L^q(\partial \Omega ))^2 \end{aligned}$$

satisfying

$$\begin{aligned} \int _{\Omega \setminus \Gamma _t}g\,\textrm{d}x=-\int _{\Gamma _{t}}\textbf{a}_{1}\cdot \textbf{n}_{\Gamma _{t}}\textrm{d}\mathcal {\mathcal {H}}^{1}+\int _{\partial \Omega }\textbf{a}\cdot \textbf{n}_{\partial \Omega }\textrm{d}\mathcal {H}^{1} \end{aligned}$$

(A.1)

for almost all $t\in (0,T)$ such that $g={\text {div}}{\textbf{G}}$ for some ${\textbf{G}}\in W^1_q(0,T;L^q(\Omega ))^2$, ${\text {div}}\textbf{v}_0|_{t=0} = g|_{t=0}$ and $\llbracket \textbf{v}_0\rrbracket =\textbf{a}_{1}|_{t=0}+ \textbf{b}_0 h_0\circ S_0|_{t=0}$, $\textbf{v}^-_0|_{\partial \Omega }= \textbf{a}|_{t=0}$, there are unique

$$\begin{aligned} \textbf{v}&\in L^q(0,T; W^2_q(\Omega \setminus \Gamma _t))^2\cap W_q^1(0,T; L^q(\Omega ))^2, \quad p\in L^q(0,T; W^1_{q,(0)}(\Omega \setminus \Gamma _t)),\\ h&\in L^q(0,T;W_q^{3-\frac{1}{q}}(\mathbb {T}^1))\cap W^1_q(0,T;W^{2-\frac{1}{q}}_q(\mathbb {T}^1)) \end{aligned}$$

solving

$$\begin{aligned} \partial _t\textbf{v}^{\pm }-\nu ^\pm \Delta \textbf{v}^{\pm }+\nabla p^{\pm }&=\textbf{f}{} & {} \text {in }\Omega ^{\pm }_t, t\in (0,T), \end{aligned}$$

(A.2)

$$\begin{aligned} {\text {div}}\textbf{v}^{\pm }&=g{} & {} \text {in }\Omega ^{\pm }_t, t\in (0,T),\end{aligned}$$

(A.3)

$$\begin{aligned} \llbracket \textbf{v}\rrbracket - \textbf{b}_0 h\circ S_0&=\textbf{a}_{1}{} & {} \text {on }\Gamma _t, t\in (0,T),\end{aligned}$$

(A.4)

$$\begin{aligned} \llbracket 2\nu D\textbf{v}-p \textbf{I}\rrbracket \textbf{n}+\overline{\nu }\llbracket \partial _{\textbf{n}} \textbf{v}\rrbracket -\sigma \Delta ^{\Gamma _t} h\circ S_0\textbf{n}&= \textbf{b}_1\partial _s h\circ S_0\nonumber \\&\quad +\textbf{b}_2h\circ S_0+ \textbf{a}_2{} & {} \text {on }\Gamma _t, t\in (0,T),\end{aligned}$$

(A.5)

$$\begin{aligned} \partial _t h+a_1\partial _s h +\tfrac{1}{2}(\textbf{v}^{+}_\textbf{n}+\textbf{v}^{-}_\textbf{n})\circ X_0 + a_0h&=w{} & {} \text {on }\mathbb {T}^1\times (0,T),\end{aligned}$$

(A.6)

$$\begin{aligned} \textbf{v}^{-}&=\textbf{a}{} & {} \text {on }\partial \Omega \times (0,T),\end{aligned}$$

(A.7)

$$\begin{aligned} \textbf{v}^\pm |_{t=0}&= \textbf{v}_0^\pm{} & {} \text {in }\Omega ,\end{aligned}$$

(A.8)

$$\begin{aligned} h|_{t=0}&= h_0{} & {} \text {in }\mathbb {T}^1, \end{aligned}$$

(A.9)

where $\textbf{v}^\pm = \textbf{v}|_{\Omega ^\pm }$, $p^\pm = p|_{\Omega ^\pm }$, $\textbf{n}=\textbf{n}_{\Gamma _t}$.

Proof

First of all by subtracting suitable extensions of $\textbf{a}$, $\textbf{v}_0$, and $h_0$ we can easily reduce to the case $\textbf{a}=0$, $\textbf{v}_0=0$, and $h_0=0$.

Step 1: Time-independent interface, zero lower order terms Let us first consider the case that $\Gamma _t=\Gamma$, $\Omega ^\pm _t =\Omega ^\pm$ are independent of $t\in [0,T_0]$, $\textbf{b}_j=0$ for $j=0,1,2$, $b_0=0$. First of all by subtracting suitable extensions, we can easily reduce to the case that $g=0$ and $\textbf{a}_1=0$. Then

$$\begin{aligned} \overline{\nu }\llbracket \nabla \textbf{v}_{\textbf{n}}\rrbracket =0\qquad \text {if } \llbracket \textbf{v}\rrbracket =0\text { on }\Gamma ,\ {\text {div}}\textbf{v}^\pm =0\text { in }\Omega ^\pm . \end{aligned}$$

Hence in this case (A.5) is equivalent to

$$\begin{aligned} \llbracket 2(\nu +\overline{\nu }) D\textbf{v}-p \textbf{I}\rrbracket \textbf{n}=\sigma \Delta ^\Gamma h\textbf{n}+\textbf{a}_2. \end{aligned}$$

(A.10)

Thus the result follows e.g. from [30, Corollary 8.1.3].

Step 2: Existence for $T=T_1>0$ sufficiently small Let $\Phi :\overline{\Omega }\times [0,T_0]\rightarrow \overline{\Omega }$ be defined by

$$\begin{aligned} \frac{d}{dt} \Phi _t(\xi )&= \textbf{v}_0^\pm (\Phi _t(\xi ),t)&\quad&\text {for all }\xi \in \overline{\Omega ^\pm _0}, t\in [0,T_0],\\ \Phi _0(\xi )&= \xi&\quad&\text {for all }\xi \in \overline{\Omega ^\pm _0}. \end{aligned}$$

Then $\Phi$ is smooth in $\overline{\Omega ^\pm }$ and $\Phi (\Gamma _0,t)=\Gamma _t$, $\Phi (\Omega ^\pm _0,t)=\Omega ^\pm _t$ for all $t\in [0,T_0]$. Moreover, $(\varvec{\textbf{v}}^\pm ,p^\pm ,h)$ solves (A.2)–(A.9) if and only if $(\tilde{\textbf{v}}^\pm ,\tilde{p}^\pm ,h)$, where $\tilde{\textbf{v}}^\pm (x,t)= \textbf{v}^\pm (\Phi _t(x),t)$, $\tilde{p}^\pm (x,t)= p^\pm (\Phi _t(x),t)$ for all $x\in \overline{\Omega _0^\pm }$, $t\in [0,T]$, solves the perturbed system

$$\begin{aligned} \partial _t\tilde{\textbf{v}}^{\pm }-\nu ^\pm \Delta \tilde{\textbf{v}}^{\pm }+\nabla \tilde{p}^{\pm }&=\tilde{\textbf{f}} + \textbf{R}_1{} & {} \text {in }\Omega ^{\pm }_0\times (0,T), \end{aligned}$$

(A.11)

$$\begin{aligned} {\text {div}}\tilde{\textbf{v}}^{\pm }&=\tilde{g} +R_2{} & {} \text {in }\Omega ^{\pm }_0\times (0,T),\end{aligned}$$

(A.12)

$$\begin{aligned} \llbracket \tilde{\textbf{v}}\rrbracket&=\tilde{\textbf{a}}_{1}+ \tilde{b}_0{\varvec{\tau }}h{} & {} \text {on }\Gamma _0\times (0,T),\end{aligned}$$

(A.13)

$$\begin{aligned} \llbracket 2\nu D\tilde{\textbf{v}}-\tilde{p} \textbf{I}\rrbracket \textbf{n}+\overline{\nu }\llbracket \partial _{\textbf{n}} \tilde{\textbf{v}}\rrbracket -\sigma \Delta ^{\Gamma _0} h\circ S_0^0&= \textbf{n}+ \tilde{\textbf{b}}_1\partial _s h\circ S_0^0\nonumber \\&\quad +\tilde{\textbf{b}}_2h\circ S_0^0+ \tilde{\textbf{a}}_2+\textbf{R}_3{} & {} \text {on }\Gamma _0\times (0,T),\end{aligned}$$

(A.14)

$$\begin{aligned} \partial _t h+a_1\partial _s h +\tfrac{1}{2}(\textbf{v}^{+}_\textbf{n}+\textbf{v}^{-}_\textbf{n})\circ X_0^0&=w- a_0h + R_4{} & {} \text {on }\mathbb {T}^1\times (0,T), \end{aligned}$$

(A.15)

together with $\tilde{\textbf{v}}|_{\partial \Omega }=0$, $\tilde{\textbf{v}}|_{t=0}=0, h|_{t=0}=0$, where $S_0^0 (x,t)= S_0(x,0)$ for all $(x,t)\in \Omega \times (0,T)$, $X_0^0(s)= X_0(s,0)$ for all $s\in \mathbb {T}^1$, and

$$\begin{aligned} \tilde{\textbf{f}}(x,t)&=\textbf{f}(\Phi _t(x),t),&\quad \tilde{g}(x,t)&=g(\Phi _t(x),t)&\qquad&\text {for all }(x,t)\in \Omega _0^\pm \times (0,T),\\ \tilde{\textbf{a}}_j(x,t)&=\textbf{a}_1(\Phi _t(x),t),&\quad \tilde{\textbf{b}}_j(x,t)&=\textbf{b}_j(\Phi _t(x),t)&\qquad&\text {for all }(x,t)\in \Gamma _0\times (0,T),j=1,2,\\ \tilde{b}_0(x,t)&=b_0(\Phi _t(x),t),{} & {} &\qquad&\text {for all }(x,t)\in \Gamma _0\times (0,T). \end{aligned}$$

Here $(\textbf{R}_1, R_2,0, \textbf{R}_3, R_4)\in \mathbb {F}(T):= \mathbb {F}_1(T)\times \ldots \times \mathbb {F}_5(T)$ depends linearly on $(\tilde{\textbf{v}}, \tilde{p},h)\in \mathbb {E}(T):= \mathbb {E}_1(T)\times \mathbb {E}_2(T) \times \mathbb {E}_3(T)$, where

$$\begin{aligned} \mathbb {E}_1(T)&:= L^q(0,T; W^2_q(\Omega \setminus \Gamma _0)\cap W^1_{q,0}(\Omega ))^2\cap {}_0W^1_q(0,T; L^q(\Omega ))^2, \\ \mathbb {E}_2(T)&:= L^q(0,T; W^1_{q,(0)}(\Omega \setminus \Gamma _0))\cap {}_0W^1_q(0,T; \dot{W}^{-1}_{q,(0)}(\Omega )) \\ \mathbb {E}_3(T)&:=L^q(0,T;W_q^{3-\frac{1}{q}}(\mathbb {T}^1))\cap {}_0 W^1_q(0,T;W_q^{2-\frac{1}{q}}(\mathbb {T}^1)),\\ \mathbb {F}_1(T)&:= L^q(\Omega \times (0,T))^2, \quad \mathbb {F}_2(T):= L^q(0,T;W^1_q(\Omega \setminus \Gamma _0)), \\ \mathbb {F}_3(T)&:=L^q(0,T;W_q^{2-\frac{1}{q}}(\Gamma _0))^2\cap {}_0W_q^{\frac{1}{2}-\frac{1}{2q}}(0,T;L^q(\Gamma _0))^2, \quad \\ \mathbb {F}_4(T)&:=L^q(0,T;W_q^{1-\frac{1}{q}}(\Gamma _0))^2\cap W_q^{\frac{1}{2}-\frac{1}{2q}}(0,T;L^q(\Gamma _0))^2, \quad \mathbb {F}_5(T):= L^q(0,T;W_q^{2-\frac{1}{q}}(\mathbb {T}^1)), \end{aligned}$$

normed in the standard way, and, for a Banach space X and $s>1-\frac{1}{q}$,

$$\begin{aligned} {}_0 W^s_q(0,T;X):=\{u \in W^s_q(0,T;X): u(0)=0\}. \end{aligned}$$

Moreover, since $\Phi _t\rightarrow _{t\rightarrow 0} {\text {id}}_{\overline{\Omega }}$ in $C^k(\overline{\Omega })$ for every $k\in \mathbb {N}$, we have that

$$\begin{aligned} \Vert (\textbf{R}_1, R_2, 0, \textbf{R}_3, {R_4})\Vert _{\mathbb {F}(T)}\le C(T) \Vert (\tilde{\textbf{v}}, \tilde{p},h)\Vert _{\mathbb {E}(T)}, \end{aligned}$$

for some $C(T)\rightarrow _{T\rightarrow 0} 0$. Furthermore using e.g. $\mathbb {E}_3(T)\hookrightarrow C([0,T];W_q^{3-\frac{3}{q}}(\mathbb {T}^1))$ one can show that

$$\begin{aligned} \Vert (0, 0, \tilde{b}_0{\varvec{\tau }}h, \tilde{\textbf{b}}_1\partial _s h\circ S_0^0+\tilde{\textbf{b}}_2h\circ S_0^0, a_0h)\Vert _{\mathbb {F}(T)}\le CT^\alpha \Vert (\tilde{\textbf{v}}, \tilde{p},h)\Vert _{\mathbb {E}(T)} \end{aligned}$$

for some $\alpha >0$ and C independent of $T\in (0,T_0]$. Hence, since the set of all invertible linear operators is open, there is some $T_1>0$ such that (A.11)–(A.15) possesses a unique solution in $(\tilde{\textbf{v}}, \tilde{p},h)\in \mathbb {E}(T)$ provided $T\in (0,T_1]$. Transforming $(\tilde{\textbf{v}},\tilde{p})$ to $\Omega ^\pm$ with the aid of $\Phi _t^{-1}$ yields the statement in this case.

Step 3: Existence for general $T>0$: Since the system is linear, the existence time $T_1>0$ in the second step is independent of the norms of the data. Moreover, as in the second step we obtain that for any $t_0\in [0,T)$ there is some $T_1(t_0)>t_0$ such that the system has a unique solution for $t\in (t_0,T_1(t_0))$ for a given initial value $\textbf{v}|_{t=t_0}=\tilde{\textbf{v}}_0$ at $t=t_0$. Because of the compactness of [0, T] and uniqueness of the solutions, we can concatenate these solutions and obtain a solution on [0, T]. $\square$

Remark A.2

With the aid of Theorem A.14 one can obtain that for all smooth $\textbf{f}, g, a_{1,2}, \textbf{b}, \textbf{w}, \textbf{a}$ (without precribed initial values $\textbf{v}_0$, $h_0$) a smooth solution of (A.2)–(A.7). To this end one extends the smooth data $\textbf{f}$, g, w, $\textbf{a}_k$, $k=1,2$, $\textbf{a}$, and $\Omega ^\pm _t, \Gamma _t$ on a time interval $[-1,T_0]$ in a smooth manner such that these functions vanish in $[-1,\frac{1}{2}]$. Then one can apply Theorem 3.11 to obtain a solution $(\textbf{v}^\pm , p^\pm ,h)$ of (A.2)–(A.7) on a time intervall $(-1,T_0)$ instead of $(0,T_0)$ and with initial values $\textbf{v}_0 =0$, $h_0=0$. Then one can apply the parameter-trick in space and time (cf. e.g. [30, Section 9.4] to obtain that $\textbf{v}^\pm ,p$ are smooth in $\bigcup _{t\in (-1,T_0]}\Omega ^\pm _t\times \{t\}$ and h is smooth in $\bigcup _{t\in (-1,T_0]}\mathbb {T}^1\times \{t\}$. Restriction to $[0,T_0]$ in time yields the existence of a smooth solution to (A.2)–(A.7), which satisfy (A.8)–(A.9) for some $\textbf{v}_0^\pm := \textbf{v}^\pm |_{t=0}$, $h_0:= h|_{t=0}$.