Least action principles for incompressible flows and geodesics between shapes

Abstract

As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to have characteristic-function densities. The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: any two shapes of equal volume can be approximately connected by an Euler spray—a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays.

Appendices

Appendix A: Some basic facts about subgradients

For the convenience of readers, we include here proofs of a few facts about subgradients that we use in Sect. 4 for the proof of Theorem 1.1. The proofs are standard and simple but seem not to be easy to extract from monographs on the subject, e.g., see [4, 10, 32].

Proposition A.1

Let H be a Hilbert space, and let $\varphi :H\rightarrow (-\infty ,\infty ]$ be convex, lower semi-continuous, and proper (i.e., somewhere finite). Let $S(x) = \frac{1}{2}\Vert x\Vert ^2+\varphi (x)$. Then:

• i. The subgradient $\partial \varphi$ is a monotone operator.

• ii. $\partial S(x) = x + \partial \varphi (x)$ for all $x\in H$.

• iii. The range of $\partial S$ is all of H. I.e., for all $y\in H$ there exists $x\in H$ and $z\in \partial \varphi (x)$ such that $y= x+z$.

Proof

i. Given any $x,{\hat{x}} \in H, z\in \partial \varphi (x)$, ${\hat{z}}\in \partial \varphi ({\hat{x}})$, by the definition of $\partial \varphi (x)$ and $\partial \varphi ({\hat{x}})$ respectively we have $\varphi ({\hat{x}}) - \varphi (x) \ge \langle z,{\hat{x}} - x \rangle$ and $\varphi (x) - \varphi ({\hat{x}}) \ge \langle {\hat{z}},x -{\hat{x}} \rangle$, whence $0\le \langle z-{\hat{z}},x-{\hat{x}} \rangle$. This proves $\partial \varphi$ is monotone.

ii. 1. Let $z\in \partial \varphi (x)$. We claim $x+z\in \partial S(x)$. Indeed, for all $h\in H$,

$$\begin{aligned} \frac{1}{2}\Vert x+h\Vert ^2 + \varphi (x+h) \ge \frac{1}{2}\Vert x\Vert ^2 + \varphi (x) + \langle z+x,h \rangle . \end{aligned}$$

2. Suppose $z\notin \partial \varphi (x)$. We claim $z+x\notin \partial S(x)$. We know there exists $h\in H$ such that

$$\begin{aligned} t^{-1}(\varphi (x+th)-\varphi (x)) - \langle z,h \rangle < 0 \end{aligned}$$

for $t=1$, hence for all $t\in (0,1]$ by convexity. Then for sufficiently small $t>0$ we can add $\frac{1}{2} t\Vert h\Vert ^2$ to the left-hand side and conclude that for small $t>0$,

$$\begin{aligned} \frac{1}{2} \Vert th\Vert ^2 + \varphi (x+th) < \varphi (x) + \langle z,th \rangle , \end{aligned}$$

whence $z+x\notin \partial S(x)$, since

$$\begin{aligned} \frac{1}{2} \Vert x+th\Vert ^2 + \varphi (x+th) < \frac{1}{2}\Vert x\Vert ^2+\varphi (x) + \langle z+x,th \rangle . \end{aligned}$$

iii. Let $y\in H$ and define ${\hat{S}}(x)=S(x)-\langle y,x \rangle =\frac{1}{2}\Vert x\Vert ^2+\varphi (x)-\langle y,x \rangle$. Due to our hypotheses, ${\hat{S}}$ has a minimum at some $x\in H$. This implies that for all $h\in H$,

$$\begin{aligned} \frac{1}{2}\Vert x+h\Vert ^2 +\varphi (x+h)\ge \frac{1}{2}\Vert x\Vert ^2 +\varphi (x)-\langle y,h \rangle , \end{aligned}$$

which means that $y\in \partial S(x)=x+\partial \varphi (x)$. $\square$

Appendix B: ${T\!L^p}$ stability of Wasserstein geodesics

Here we recall the notion of ${T\!L^p}$ convergence as introduced in [28], which provides a more precise way to compare Wasserstein geodesics than the notion of weak convergence does alone. We establish the ${T\!L^p}$ stability of optimal transport maps in Theorem B.1 and ${T\!L^p}$ stability of the Wasserstein geodesics in Corollary B.2. In Proposition B.4 we recall a basic property of ${T\!L^p}$ convergence and use it to show that the stability in Corollary B.2 holds even if the maps used to couple the relevant measures are not optimal. This technical result is needed in the proofs in Sect. 6.2.

The ${T\!L^p}$ metric provides a natural setting for comparing optimal transport maps between different probability measures. Let ${{\mathcal {P}}}_p({\mathbb {R}}^d)$ be the space of probability measures on ${\mathbb {R}}^d$ with finite p-th moments. On the space ${T\!L^p}({\mathbb {R}}^d)$, consisting of all ordered pairs $(\mu ,g)$ where $\mu \in {{\mathcal {P}}}_p({\mathbb {R}}^d)$ and $g \in L^p(\mu )$, the metric is given as follows: For $1\le p<\infty$,

$$\begin{aligned} d_{{T\!L^p}} (( \mu _0, g_0), ( \mu _1, g_1)) = \inf _{\pi \in \Pi ( \mu _0, \mu _1)}\left( \int |x-y|^p + | g_0(x) - g_1(y)|^p\ d \pi (x,y) \right) ^{1/p},\\ d_{T\!L^\infty }((\mu _0,g_0),(\mu _1,g_1)) = \inf _{\pi \in \Pi ( \mu _0, \mu _1)} \mathop {\mathrm{ess\, sup}}_\pi (|x-y|+|g_0(x)-g_0(y)| ) , \end{aligned}$$

where $\Pi ( \mu _0, \mu _1)$ is the set of transportation plans (couplings) between $\mu _0$ and $\mu _1$.

The following result establishes a (new) ${T\!L^p}$ stability property for optimal transport maps, as a consequence of a known general stability property for optimal plans.

Theorem B.1

(${T\!L^p}$ stability of transport maps) Let $\mu ,\mu _k\in {{\mathcal {P}}}_p({\mathbb {R}}^d)$ be probability measures absolutely continuous with respect to Lebesgue measure, and let $\nu ,\nu _k\in {{\mathcal {P}}}_p({\mathbb {R}}^d)$, for each $k\in {\mathbb {N}}$. Assume that

$$\begin{aligned} d_p(\mu _k, \mu ) \rightarrow 0\; \text { and } \; d_p(\nu _k,\nu ) \rightarrow 0 \; \text { as } k \rightarrow \infty . \end{aligned}$$

Let $T_k$ and T be the optimal transportation maps between $\mu _k$ and $\nu _k$, and $\mu$ and $\nu$, respectively. Then

$$\begin{aligned} (\mu _k,T_k) \overset{{T\!L^p}}{\longrightarrow } (\mu ,T) \text { as } k \rightarrow \infty . \end{aligned}$$

Proof

The measures $\pi _k= ({{\,\mathrm{id}\,}}\times T_k)_\sharp \mu _k$ and $\pi = ({{\,\mathrm{id}\,}}\times T)_\sharp \mu$ are the optimal transportation plans between $\mu _k$ and $\nu _k$, and $\mu$ and $\nu$, respectively. By stability of optimal transport plans (Proposition 7.1.3 of [2] or Theorem 5.20 in [53]) the sequence $\pi _k$ is precompact with respect to weak convergence and each of its subsequential limits is an optimal transport plan between $\mu$ and $\nu$. Since $\pi$ is the unique optimal transportation plan between $\mu$ and $\nu$ the sequence $\pi _k$ converges to $\pi$. Furthermore, by Theorem 5.11 of [47] or Remark 7.1.11 of [2],

$$\begin{aligned} \lim _{k \rightarrow \infty } \int |x|^p + |y|^p \ d \pi _k(x,y)&= \lim _{k \rightarrow \infty } \int |x|^p\, d\mu _k + \int |y|^p\, d \nu _k \\&= \int |x|^p\, d\mu + \int |y|^p\, d \nu = \int |x|^p + |y|^p\ d \pi (x,y). \end{aligned}$$

By Lemma 5.1.7 of [2], it follows the $\pi _k$ have uniformly integrable p-th moments, therefore

$$\begin{aligned} d_p(\pi _k, \pi )\rightarrow 0 \quad \text{ as } k \rightarrow \infty , \end{aligned}$$

by Proposition 7.1.5 in [2]. Hence there exists (optimal) $\gamma _k \in \Pi (\pi , \pi _k)$ such that

$$\begin{aligned} \int |x - {\tilde{x}}|^p + |y- {\tilde{y}}|^p \ d\gamma _k(x,y,{\tilde{x}}, {\tilde{y}}) \rightarrow 0 \quad \text { as } k \rightarrow \infty . \end{aligned}$$

(B.1)

Since $\pi$-almost everywhere $y = T(x)$ and $\pi _k$-almost everywhere ${\tilde{y}} = T_k({\tilde{x}})$ and the support ${{\,\mathrm{supp}\,}}\gamma _k$ of $\gamma _k$ is contained in ${{\,\mathrm{supp}\,}}\pi \times {{\,\mathrm{supp}\,}}\pi _k$, we conclude that $\gamma _k$-almost everywhere $(x,y, {\tilde{x}}, {\tilde{y}}) = (x,T(x), {\tilde{x}}, T_k({\tilde{x}}))$. Therefore

$$\begin{aligned} \int |x - {\tilde{x}}|^p + |T(x)- T_k({\tilde{x}})|^p\ d\gamma _k(x,y,{\tilde{x}}, {\tilde{y}}) \rightarrow 0 \quad \text { as } k \rightarrow \infty . \end{aligned}$$

Finally let $\theta _k$ be the projection of $\gamma _k$ to $(x, {\tilde{x}})$ variables. Since $\theta _k \in \Pi (\mu , \mu _k)$, by above

$$\begin{aligned} \int |x - {\tilde{x}}|^p + |T(x)- T_k({\tilde{x}})|^p\ d\theta _k(x, {\tilde{x}}) \rightarrow 0 \quad \text { as } k \rightarrow \infty . \end{aligned}$$

(B.2)

Thus $(\mu _k, T_k) \overset{{T\!L^p}}{\longrightarrow } (\mu ,T)$. $\square$

We now consider the convergence of Wasserstein geodesics between measures $\mu _k$ and $\nu _k$ as in the Lemma B.1, treating only the case $p=2$. We recall that particle paths along these geodesics are given by

$$\begin{aligned} T_{k,t}(x) = (1-t)x + t T_k(x). \end{aligned}$$

The displacement interpolation between $\mu _k$ and $\nu _k$, and particle velocities (in Eulerian variables) along the geodesics, are given by (cf. (6.15)–(6.16))

$$\begin{aligned} \mu _{k,t} = {T_{k,t}}_\sharp \mu _k, \quad v_{k,t}= (T_k-{{\,\mathrm{id}\,}})\circ T_{k,t}^{-1}, \quad t\in [0,1). \end{aligned}$$

If $\nu _k$ is absolutely continuous with respect to Lebesgue measure, then $t=1$ is allowed. We also recall that

$$\begin{aligned} \int |v_{k,t}(z)|^2 d \mu _{k,t}(z) = \int |v_{k,0}(x)|^2 d \mu _k(x) = d_2^2(\mu _k, \nu _k). \end{aligned}$$

Furthermore it is straightforward to check that $t \mapsto (\mu _{k,t}, v_{k,t})$ is Lipschitz continuous into $T\!L^2({\mathbb {R}}^d)$, satisfying for $0\le s<t<1$

$$\begin{aligned} (t-s)d_2(\mu _k,\nu _k)=d_2(\mu _{k,t},\mu _{k,s}) \le d_{{T\!L^2}}((\mu _{k,t},v_{k,t}),(\mu _{k,s},v_{k,s})) \le (t-s)d_2(\mu _k,\nu _k).\nonumber \\ \end{aligned}$$

(B.3)

Corollary B.2

(${T\!L^2}$ stability for displacement interpolants) Under the assumptions of Theorem B.1 for the case $p=2$, as $k\rightarrow \infty$ we have

$$\begin{aligned} \sup _{t\in [0,1]} d_2(\mu _{k,t},\mu _t) \rightarrow 0 \quad \text{ and }\quad \sup _{t\in [0,1)} d_{{T\!L^2}}( (\mu _{k,t},v_{k,t}),(\mu _t,v_t)) \rightarrow 0. \end{aligned}$$

(B.4)

If the measures $\nu _k$ and $\nu$ are absolutely continuous with respect to Lebesgue measure then the convergence in (B.4) also holds for $t\in [0,1]$.

Proof

Let $\pi \in \Pi (\mu , \nu ), \pi _k \in \Pi (\mu _k, \nu _k)$, and $\gamma _k \in \Pi ( \pi , \pi _k)$ be as in the proof of Theorem B.1. Similarly to $\theta _k$, we define $\theta _{k,t} = (z_t \times z_t)_{\sharp } \gamma _k$ where

$$\begin{aligned} z_t(x,y) = (1-t)x+ty \quad \text{ and }\quad (z_t \times z_t)(x,y,{\tilde{x}}, {\tilde{y}}) = (z_t(x,y), z_t({\tilde{x}}, {\tilde{y}})). \end{aligned}$$

We note that $\theta _{k,t} \in \Pi (\mu _t, \mu _{k,t})$ and hence, for all $t\in [0,1]$,

$$\begin{aligned} d_2(\mu _t, \mu _{k,t})^2&\le \int |z-{\tilde{z}}|^2 d \theta _{k,t}(z,{\tilde{z}}) \\&= \int |(1-t)(x-{\tilde{x}}) + t(y - {\tilde{y}})|^2 d \gamma _k(x, y, {\tilde{x}}, {\tilde{y}}) \\&\le 2 \int |x - {\tilde{x}}|^2 + |y - {\tilde{y}}|^2 d \gamma _k(x, y, {\tilde{x}}, {\tilde{y}}) , \end{aligned}$$

which by (B.1) converges to 0 as $k \rightarrow \infty$.

We use the same coupling $\theta _{k,t}$ to compare the velocities. Using that $\gamma _k$-almost everywhere $(x,y, {\tilde{x}}, {\tilde{y}}) = (x,T(x), {\tilde{x}}, T_k({\tilde{x}}))$, for any $t\in [0,1)$ we obtain

$$\begin{aligned}&\int |v_t(z) - v_{k,t}({\tilde{z}})|^2\ d \theta _{k,t}(z, {\tilde{z}}) \\&\qquad = \int |v_t((1-t)x + ty) - v_{k,t}((1-t) {\tilde{x}} + t {\tilde{y}})|^2\ d \gamma _k(x, y, {\tilde{x}}, {\tilde{y}}) \\&\qquad = \int |v_t(T_t(x)) - v_{k,t}(T_{k,t}({\tilde{x}}))|^2\ d \theta _k(x, {\tilde{x}}) \\&\qquad = \int |v_0(x)-v_{k,0}({\tilde{x}})|^2 \ d \theta _k(x, {\tilde{x}}) \\&\qquad \le 2 \int |x - {\tilde{x}}|^2 + |T(x) - T_k({\tilde{x}})|^2 \ d \theta _k(x, {\tilde{x}}), \end{aligned}$$

which converges to 0 as $k \rightarrow \infty$, as in (B.2). $\square$

Remark B.3

If the target measure $\nu _k$ is not absolutely continuous with respect to Lebesgue measure, then $T_k$ may fail to be invertible on the support of $\nu _k$ and $(\mu _{k,t},v_{k,t})$ may fail to converge as $t\rightarrow 1$ to some point in ${T\!L^2}({\mathbb {R}}^d)$ due to oscillations in velocity. However, if $\nu _k$ and $\nu$ are absolutely continuous with respect to Lebesgue measure, then the curves $t \mapsto (\mu _{k,t}, v_{k,t}), t\mapsto (\mu _t,v_t)$ extend as continuous maps into ${T\!L^2}$ for all $t\in [0,1]$, and the uniform convergences in (B.4) holds on [0, 1].

A number of properties of the ${T\!L^p}$ metric are established in Section 3 of [28] for measures supported in a fixed bounded set. One useful characterization of ${T\!L^p}$-convergence in this case is stated in Proposition 3.12 of [28], which implies the following.

Proposition B.4

(A characterization of ${T\!L^p}$ convergence on bounded domains) Let $D\subset {\mathbb {R}}^d$ be open and bounded, and let $\mu$ and $\mu _k$ ($k\in {\mathbb {N}}$) be probability measures on D, and suppose $\mu$ is absolutely continuous with respect to Lebesgue measure. Call a sequence of transport maps $(S_k)$ that push forward $\mu$ to $\mu _k$ (satisfying ${S_k}_\sharp \mu =\mu _k$) stagnating if

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _D |x-S_k(x)|\,d\mu (x) = 0 . \end{aligned}$$

(B.5)

Then the following are equivalent, for $1\le p<\infty$.

• (i) $(\mu _k,f_k)\overset{{T\!L^p}}{\longrightarrow } (\mu ,f)$ as $k\rightarrow \infty$.

• (ii) $\mu _k$ converges weakly to $\mu$ and there exists a stagnating sequence $(S_k)$ such that

  $$\begin{aligned} \int _D |f(x)-f_k(S_k(x))|^p \,d\mu (x) \rightarrow 0 \quad \text{ as } k\rightarrow \infty . \end{aligned}$$

  (B.6)

• (iii) $\mu _k$ converges weakly to $\mu$ and for every stagnating sequence $(S_k)$ the equality (B.6) holds.

This result together with Proposition B.2 yields the following.

Corollary B.5

(A characterization of ${T\!L^p}$ convergence for displacement interpolants) Make the same assumptions as in Corollary B.2, and assume all measures $\mu _k, \mu$, $\nu _k, \nu$ are absolutely continuous with respect to Lebesgue measure and have support in a bounded open set D. Then for any stagnating sequence of transport maps $(S_k)$ that push forward $\mu$ to $\mu _k$, with the notation

$$\begin{aligned} S_{k,t} = T_{k,t}\circ S_k\circ T_t^{-1}\end{aligned}$$

the sequence $(S_{k,t})$ pushes forward $\mu _t$ to $\mu _{k,t}$ and is stagnating, and as $k\rightarrow \infty$,

$$\begin{aligned}&\sup _{t\in [0,1]} \int |x-S_{k,t}(x)|^2\,d\mu _t(x) \rightarrow 0, \end{aligned}$$

(B.7)

$$\begin{aligned}&\sup _{t\in [0,1]} \int |v_t(x)-v_{k,t}(S_{k,t}(x))|^2\,d\mu _t(x) \rightarrow 0, \end{aligned}$$

(B.8)

Proof

First we note that indeed

$$\begin{aligned} \mu _{k,t}=(T_{k,t})_\sharp \mu _k = (T_{k,t}\circ S_k)_\sharp \mu = (S_{k,t})_\sharp \mu _t. \end{aligned}$$

Next, fix any $t\in [0,1]$. Because $d_2(\mu _{k,t},\mu _t)\rightarrow 0$ by (B.4) and $T_{k,t}$ is the optimal transport map pushing forward $\mu _k$ to $\mu _{k,t}$, by Theorem B.1 we have $d_2( (\mu _k,T_{k,t}),(\mu ,T_t))\rightarrow 0$. Now by Proposition B.4, because $(T_t)_\sharp \mu =\mu _t$ we have

$$\begin{aligned} \int |x-S_{k,t}(x)|^2 \,d\mu _t(x) = \int |T_t(z)-T_{k,t}(S_k(z))|^2\,d\mu (z)\rightarrow 0. \end{aligned}$$

(B.9)

We infer that $(S_{k,t})$ is stagnating and the convergence in (B.7) holds pointwise in t. But now, the middle quantity in (B.9) is a quadratic function of t, so the uniform convergence in (B.7) holds.

Next, we note that the quantity in (B.8) is actually independent of t. We have

$$\begin{aligned} \int |v_t(x)-v_{k,t}(S_{k,t}(x))|^2\,d\mu _t(x) =\int |v_0(z)-v_{k,0}(S_k(z))|^2\,d\mu (z) \rightarrow 0, \end{aligned}$$

due to Proposition B.4. $\square$