On local well-posedness of the thin-film equation via the Wasserstein gradient flow

Abstract

A local existence and uniquness of the gradient flow of one dimensional Dirichlet energy on the Wasserstein space is proved. The proofs are based on a relaxation of displacement convexity in the Wasserstein space and can be applied to a family of higher order energy functionals which are not displacement convex in the standard sense. As the result a local well-posedness of the corresponding nonlinear evolution equations including the thin-film equation and the quantum drift diffusion equation are proved.

Appendix

In this part we recall some basic elements of the Riemannian structure of the Wasserstein space $\mathcal {P}_2({\mathbb {R}}^m)$ very briefly. We refer the reader to [1] or [12] for detail discussions and proofs.

Lets start by the definition of a gradient flow of an energy on the Wasserstein space. Assume that an energy $E$ is defined on $\mathcal {P}_2({\mathbb {R}}^m)$. We say that an absolutely continuous curve $\mu _t$ is a trajectory of the gradient flow for the energy $E$, if there exists a velocity field $V_t$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\mu _t + \nabla \cdot \left( \mu _t V_t \right) =0 &{}\quad \text {(continuity equation)}, \\ V_t \in -\partial E(\mu _t) &{}\quad \text {(steepest descent)} , \end{array}\right. } \end{aligned}$$

(26)

hold for almost every $t>0$. We refer to (26) as the gradient flow equation. The continuity equation, which holds in distributional sense, links the curve with its velocity vector field and it also ensures that the mass is conserved. The steepest descent equation expresses that the gradient flow evolves in the direction of maximal energy dissipation.

Consider the Wasserstein distance (1). The Brenier-McCann theorem [4] asserts that if $\mu \in \mathcal {P}_2^a({\mathbb {R}}^m)$ where $\mathcal {P}_2^a({\mathbb {R}}^m)$ is the set of absolutely continuous probability measures with respect to the Lebesgue measure, then the Wasserstein distance simplifies to

$$\begin{aligned} W_2(\mu ,\nu )=\left( \int \limits _{{\mathbb {R}}^m} \left| T_{\mu }^{\nu }-Id \right| ^2 d\mu \right) ^{\frac{1}{2}}, \end{aligned}$$

(27)

where the map $T_{\mu }^{\nu }$ is called the optimal map. Assuming $\mu =udx$ and $\nu =vdx$ are absolutely continuous measures, the MongeAmp equation gives an explicit relation between the densities $u$ and $v$ in terms of the optimal map. For a.e. $x$ we have

$$\begin{aligned} v(T_{\mu }^{\nu }(x))=\frac{u(x)}{det(DT_{\mu }^{\nu })(x)}. \end{aligned}$$

(28)

The optimal map $T_{\mu }^{\nu }(x)$ also defines a geodesic $\mu _s$ between two measures $\mu _0$ and $\mu _1$

$$\begin{aligned} \mu _s=\left( (1-s)Id + sT_{\mu _0}^{\mu _1} \right) _{\#}\mu _0. \end{aligned}$$

(29)

In [2] Brenier and Benamou showed that the Wasserstein space is a length space in the sense that

$$\begin{aligned} W_2(\mu ,\nu )=\inf \left\{ \int \limits _0^1 \left( \int _{{\mathbb {R}}^m}|V_t|^2d\mu _t \right) ^{\frac{1}{2}} dt ~~\text {s.t.}~~ \partial _t\mu _t + \nabla . (\mu _t V_t) =0; ~ \mu _0=\mu ,~\mu _1=\nu \right\} \end{aligned}$$

(30)

where the infimum is taken over all curves in $AC^2([0,1]; \mathcal {P}_2({\mathbb {R}}^m))$. For such a given curve $\mu _t$, there is a unique vector field $V_t$ that attains the minimum. This optimal velocity vector field is defined to be the tangent vector field to the curve $\mu _t$. The tangent vector field of $\mu _t$ can also be expressed in term of the optimal maps. If $V_t$ is the tangent vector field of $\mu _t$ then for almost every $t$ we have

$$\begin{aligned} V_t=\lim _{\epsilon \rightarrow 0} \dfrac{T_{\mu _t}^{\mu _{t+\epsilon }}-Id}{\epsilon }. \end{aligned}$$

Furthermore, by [1, Chapter 8] the derivative of the Wasserstein metric along $\mu _t$ is given by

$$\begin{aligned} \dfrac{d}{dt}W_2(\mu _t,\nu )^2= 2 \int \limits _{{\mathbb {R}}^m}\langle V_t, Id - T_{\mu _t}^{\nu } \rangle d\mu _t\quad ~~~~ \forall \nu \in \mathcal {P}_2({\mathbb {R}}^m) \end{aligned}$$

(31)

for a.e. $t$. A vector field $\xi \in L^2(d\mu )$ belongs to the subdifferential of $E$ at $\mu$ if

$$\begin{aligned} \liminf _{\begin{array}{c} \nu \rightarrow \mu \\ \nu \in D(E) \end{array}} \dfrac{E(\nu )-E(\mu )- \int \limits _X \left\langle \xi , T_{\mu }^{\nu }-Id \right\rangle \,\mathrm {d}\mu }{W_2\left( \mu ,\nu \right) } \geqslant 0. \end{aligned}$$

(32)

The Wasserstein metric is closely related to a certain weak topology on $\mathcal{P}_2({\mathbb {R}}^m)$. This topology is induced by narrow convergence:

$$\begin{aligned} \mu _n \xrightarrow {\text {narrow}} \mu ~~\Longleftrightarrow ~~ \int \limits _{{\mathbb {R}}^m} fd\mu _n \rightarrow \int \limits _{{\mathbb {R}}^m} fd\mu ~~~~\quad \forall f\in C_b^0({\mathbb {R}}^m)\,. \end{aligned}$$

(33)

The topologies induced by the narrow convergence and the Wasserstein distance are equivalent for sequences of measures with uniformly bounded second moments:

$$\begin{aligned} \lim _{n \rightarrow \infty } W_2(\mu _n,\mu )=0\,\,\Longleftrightarrow {\left\{ \begin{array}{ll} \mu _n \xrightarrow {\text {narrow}} \mu \\ \lbrace \mu _n \rbrace \,\,\text {has uniformly bounded 2-moments}.\\ \end{array}\right. } \end{aligned}$$

(34)

In the formal level, the link between Wasserstein gradient flows and evolution PDEs is described as follows. assuming that $E(\mu _t)$ is smooth in $t$ then $\mu _t$ is trajectory of a Wasserstein gradient flow of $E$ if

$$\begin{aligned} \partial _t u = \nabla . \left( u \nabla \left( \dfrac{\delta E(u)}{\delta u}\right) \right) \end{aligned}$$

(35)

where $\frac{\delta E(u)}{\delta u}$ is the standard first variation of $E$. For example in the case of the Dirichlet energy $E(u)=\int _{{\mathbb {R}}^m}|\nabla u|^2 dx$, the PDE corresponding to its Wasserstein gradient flow thin-film equation $\partial _t u = - \nabla .(u \nabla \Delta u)$.

The notion of minimizing movement scheme as a discrete-time approximation of a gradient flow is very useful. Let $\mu _0 \in D(E)$, and fix the step size $\tau >0$. Recursively define a sequence $\left\{ M_{\tau }^n \right\} _{n=1}^{+\infty }$ by setting $M_0^\tau =\mu _0$, and for $n\geqslant 1$,

$$\begin{aligned} M_{n}^{\tau } = \underset{\mu \in D(E)}{\mathrm{argmin}} \left\{ E (\mu ) + \dfrac{1}{2\tau } W_2^2 \left( M_{n-1}^{\tau },\mu \right) \right\} . \end{aligned}$$

(36)

Next, define a piecewise constant curve and a corresponding velocity field by

$$\begin{aligned} \mu ^\tau _t:= M_n^{\tau }\,,\qquad V^{\tau }_t:=-\frac{T_{M^\tau _{n}}^{M^\tau _{n-1}}-Id}{\tau }\,,\qquad \text{ for }\ (n-1)\tau <t\le n\tau \,. \end{aligned}$$

The formal Euler-Lagrange equation for this minimization problem is given by

$$\begin{aligned} V^\tau _t \in -\partial E(\mu ^\tau _t ) \,\,\,\,\quad \forall t>0. \end{aligned}$$

(37)

This equation suggests that $\mu _t^{\tau }$ is an approximation of the gradient flow trajectory of $E$ starting from $\mu _0$.

An energy functional $E$ is called $\varvec{\lambda }$ -convex along $\mu _t$ if the convexity is bounded from below by the constant $\lambda$, i.e.

$$\begin{aligned} E(\mu _s)\leqslant (1-s)E(\mu _0)+sE(\mu _1)-\dfrac{\lambda }{2}s(1-s)W_2^2(\mu _0,\mu _1)\,\,\,\quad s \in [0,1]. \end{aligned}$$

(38)

along all geodesics of $\mathcal {P}_2({\mathbb {R}}^m)$. Furthermore, assuming that $E(\mu _s)$ is smooth in $s$, one can write a derivative version of $\lambda$-convexity. In this case, $E$ is $\lambda$-convex if

$$\begin{aligned} \dfrac{d^2}{ds^2}E(\mu _s)\geqslant \lambda W_2^2(\mu _0,\mu _1). \end{aligned}$$

(39)

Under $\lambda$-convexity assumption, the gradient flow of an energy exists and is unique (see [1]).