On nonlinear cross-diffusion systems: an optimal transport approach

Abstract

We study a nonlinear, degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework is introduced based on its gradient flow structure in Wasserstein space to derive a notion of discrete-time solutions. Its continuum limit, due to the possible mixing of the densities, only solves a weaker version of the original system. In one space dimension, we find a stable initial configuration which allows the densities to be segregated. This leads to the evolution of a stable interface between the two densities, and to a stronger convergence result to the continuum limit. In particular derivation of a standard weak solution to the system is available. We also study the incompressible limit of the system, which addresses transport under a height constraint on the total density. In one space dimension we show that the problem leads to a two-phase Hele-Shaw type flow.

Appendices

Appendix A: Optimal transport toolbox

Lemma A.1

Let $f:[0,+\,\infty )\rightarrow {\mathbb {R}}$ be a $C^1$ convex function that is superlinear at $+\,\infty .$ Let $M>0.$ We consider ${\mathcal {F}}:{\mathscr {P}}^M(\Omega )\rightarrow {\mathbb {R}}\cup {+\,\infty }$ defined as

$$\begin{aligned} {\mathcal {F}}(\rho )=\left\{ \begin{array}{ll} \displaystyle \int _\Omega f(\rho (x))\mathrm{d}x, &{}\quad {\mathrm{if}}\ \rho \ll \mathscr {L}^d,\\ +\,\infty , &{}\quad {\mathrm{otherwise}}. \end{array}\right. \end{aligned}$$

Let $\nu \in {\mathscr {P}}^{M}(\Omega )$ be given. Then there exists a solution $\varrho \in {\mathscr {P}}^{{\mathrm{ac}},M}(\Omega )$ of the minimization problem

$$\begin{aligned} \min _{\rho \in {\mathscr {P}}^M(\Omega )}\left\{ {\mathcal {F}}(\rho )+\frac{1}{2} W_2^2(\rho ,\nu )\right\} . \end{aligned}$$

If in addition $\nu \ll \mathscr {L}^d$ or if f is strictly convex, then $\varrho$ is unique.

Moreover, $\exists C\in {\mathbb {R}}$ such that for a suitable Kantorovich potential $\varphi$ in the optimal transport of $\varrho$ onto $\nu$ one has the following first order necessary optimality condition fulfilled

$$\begin{aligned} \left\{ \begin{array}{ll} f'(\varrho )+\varphi =C, &{}\quad \varrho -{\mathrm{a.e.}},\\ f'(\varrho )+\varphi \ge C, &{}\quad {\mathrm{on}}\ \{\varrho =0\}. \end{array}\right. \end{aligned}$$

(A.1)

If $f'(0)$ is finite, then one can express the above condition as $f'(\varrho )=\max \{C-\varphi , f'(0)\}.$

Proof

The proof of the previous results can be found in [11] or [43, Chapter 7]. $\square$

It turns out that $({\mathscr {P}}^M(\Omega ),W_2)$ is a geodesic space and constant speed geodesics (and absolutely continuous curves in general) can be characterized by special solutions of continuity equations. Since this characterization is true for any $M>0,$ we simply set $M=1$ in the theorem below.

Theorem A.2

(see [3, 43])

• 1. Let $\Omega \subset {\mathbb {R}}^d$ compact and $(\mu _t)_{t\in [0,T]}$ be an absolutely continuous curve in $({\mathscr {P}}(\Omega ),W_2).$ Then for a.e. $t\in [0,T]$ there exists a vector field $\mathbf{v }_t\in L^2_{\mu _t}(\Omega ;{\mathbb {R}}^d)$ s.t.

  • the continuity equation $\partial _t\mu _t+\nabla \cdot (\mathbf{v }_t\mu _t)=0$ is satisfied in the weak sense;

  • for a.e. $t\in [0,T],$ one has $\Vert v_t\Vert _{L^2_{\mu _t}}\le |\mu '|_{W_2}(t),$ where

    $$\begin{aligned} |\mu '|_{W_2}(t):=\lim _{h\rightarrow 0}\frac{W_2(\mu _{t+h},\mu _t)}{|h|} \end{aligned}$$

    denotes the metric derivative of the curve $[0,T]\ni t\mapsto \mu _t$ w.r.t. $W_2,$ provided the limit exists.

• 2. Conversely, if $(\mu _t)_{t\in [0,T]}$ is a family of measures in ${\mathscr {P}}(\Omega )$ and for each t one has a vector field $\mathbf{v }_t\in L^2_{\mu _t}(\Omega ;{\mathbb {R}}^d)$ s.t. $\int _0^T\Vert \mathbf{v }_t\Vert _{L^2_{\mu _t}}\mathrm{d}t<+\,\infty$ and $\partial _t\mu _t+\nabla \cdot (\mathbf{v }_t\mu _t)=0$ in the weak sense, then $[0,T]\ni t\mapsto \mu _t$ is an absolutely continuous curve in $({\mathscr {P}}(\Omega ),W_2)$, with $|\mu '|_{W_2}(t)\le \Vert v_t\Vert _{L^2_{\mu _t}}$ for a.e. $t\in [0,T]$ and $W_2(\mu _{t_1},\mu _{t_2})\le \int _{t_1}^{t_2} |\mu '|_{W_2}(t)\mathrm{d}t.$ If moreover $|\mu '|\in L^2(0,T)$, then we say that $\mu$ belongs to the space $AC^2([0,T];({\mathscr {P}}(\Omega ), W_2)).$

• 3. For curves $(\mu _t)_{t\in [0,1]}$ that are geodesics in $({\mathscr {P}}(\Omega ),W_2)$ one has the equality

  $$\begin{aligned} W_2(\mu _0,\mu _1)=\int _0^1 |\mu '|_{W_2}(t)\mathrm{d}t=\int _0^1 \Vert v_t\Vert _{L^2_{\mu _t}}\mathrm{d}t. \end{aligned}$$

• 4. For $\mu _0,\mu _1\in {\mathscr {P}}^{{\mathrm{ac}}}(\Omega )$, a constant speed geodesic connecting them is a curve $(\mu _t)_{t\in [0,1]}$ such that $W_2(\mu _s,\mu _t)=|t-s|W_2(\mu _0,\mu _1)$ for any $t,s\in [0,1].$ One can compute this constant speed geodesic using McCann’s interpolation, i.e. $\mu _t:=\left( T_t\right) _\#\mu _0,$ for all $t\in [0,1]$, where $T_t:=(1-t)\mathrm{id}+tT$ with $T_\#\mu _0=\mu _1$ the optimal transport map between $\mu _0$ and $\mu _1$. Moreover, the velocity field in the continuity equation is given by $\mathbf{v }_t:=(T-\mathrm{id})\circ (T_t)^{-1}.$

Let us introduce the Benamou-Brenier functional ${\mathcal {B}}_2:{\mathscr {M}}([0,T]\times \Omega )\times {\mathscr {M}}^d([0,T]\times \Omega )\rightarrow {\mathbb {R}}\cup \{+\,\infty \}$ defined as

$$\begin{aligned} {\mathcal {B}}_2(\mu , \mathbf{E }):=\left\{ \begin{array}{ll} \displaystyle \int _0^T\int _\Omega |\mathbf{v }_t|^2\mathrm{d}\mu _t(x)\mathrm{d}t, &{}\quad \text {if}\ \mathbf{E }=\mathbf{E }_t\otimes \mathrm{d}t, \mu =\mu _t\otimes \mathrm{d}t \ \text {and}\ \mathbf{E }_t=\mathbf{v }_t\cdot \mu _t,\\ \displaystyle +\,\infty , &{}\quad \text {otherwise}. \end{array} \right. \end{aligned}$$

It is well-known (see for instance [43, Proposition 5.18]) that ${\mathcal {B}}_2$ is jointly convex and lower semicontinuous w.r.t. the weak$-\star$ convergence. In particular if $(\mu ,\mathbf{E })$ solves $\partial _t\mu +\nabla \cdot \mathbf{E }=0$ in the weak sense with ${\mathcal {B}}_2(\mu ,\mathbf{E })<+\,\infty ,$ implies that $t\mapsto \mu _t$ is a curve in $AC^2([0,T];({\mathscr {P}}(\Omega ), W_2)).$

The following comparison result appears to be well-known but we write it here for completeness.

Lemma A.3

Let $\mu ^1,\nu ^1\in {\mathscr {P}}^{M_1}(\Omega )$ and $\mu ^2,\nu ^2\in {\mathscr {P}}^{M_2}(\Omega ).$ Then the following inequality holds true

$$\begin{aligned} W_2^2(\mu ^1+\mu ^2,\nu ^1+\nu ^2)\le W_2^2(\mu ^1,\nu ^1) + W_2^2(\mu ^2,\nu ^2). \end{aligned}$$

(A.2)

Remark A.1

Note that with the abuse of notation, $W_2$ on the l.h.s. of (A.2) denotes the $2-$Wasserstein distance on ${\mathscr {P}}^{M_1+M_2}(\Omega ),$ while on the r.h.s. $W_2$ denotes the corresponding distances on ${\mathscr {P}}^{M_1}(\Omega )$ and ${\mathscr {P}}^{M_2}(\Omega )$ respectively.

Proof of Lemma A.3

The quantity on the l.h.s. of (A.2) is realized by an optimal plan $\gamma \in \Pi ^{M_1+M_2}(\mu ^1+\mu ^2,\nu ^1+\nu ^2)$ i.e.

$$\begin{aligned} W_2^2(\mu ^1+\mu ^2,\nu ^1+\nu ^2)=\int _{\Omega \times \Omega }|x-y|^2\mathrm{d}\gamma . \end{aligned}$$

Similarly the quantities on the r.h.s. can be written with the help of some optimal plans $\gamma ^i\in \Pi ^{M_i}(\mu ^i,\nu ^i),$ $i=1,2,$ i.e.

$$\begin{aligned} W_2^2(\mu ^i,\nu ^i)=\int _{\Omega \times \Omega }|x-y|^2\mathrm{d}\gamma ^i. \end{aligned}$$

Now set ${\tilde{\gamma }}:=\gamma ^1+\gamma ^2$. Clearly since $(\pi ^x)_\#{\tilde{\gamma }}=\mu ^1+\nu ^1$ and $(\pi ^y)_\#{\tilde{\gamma }}=\mu ^2+\nu ^2$ one has ${\tilde{\gamma }}\in \Pi ^{M_1+M_2}(\mu ^1+\mu ^2,\nu ^1+\nu ^2)$. Hence

$$\begin{aligned} W_2^2(\mu ^1+\mu ^2,\nu ^1+\nu ^2)&=\int _{\Omega \times \Omega }|x-y|^2\mathrm{d}\gamma \le \int _{\Omega \times \Omega }|x-y|^2\mathrm{d}{\tilde{\gamma }}\\&=\int _{\Omega \times \Omega }|x-y|^2\mathrm{d}\gamma ^1+\int _{\Omega \times \Omega }|x-y|^2\mathrm{d}\gamma ^2\\&\le W_2^2(\mu ^1,\nu ^1)+W_2^2(\mu ^2,\nu ^2). \end{aligned}$$

Therefore, inequality (A.2) follows. $\square$

Appendix B: A refined Aubin–Lions lemma

In [40] the authors present the following version of the classical Aubin–Lions lemma (see [4]):

Theorem B.1

[40, Theorem 2] Let B be a Banach space and ${\mathcal {U}}$ be a family of measurable B-valued function. Let us suppose that there exist a normal coercive integrand ${\mathfrak {F}}:(0,T)\times B\rightarrow [0,+\,\infty ]$, meaning that

• 1. ${\mathfrak {F}}$ is $\mathscr {B}(0,T)\otimes \mathscr {B}(B)$-measurable, where $\mathscr {B}(0,T)$ and $\mathscr {B}(B)$ denote the $\sigma$-algebgras of the Lebesgue measurable subsets of (0, T) and of the Borel subsets of B respectively;

• 2. the maps $v\mapsto {\mathfrak {F}}_t(v):={\mathfrak {F}}(t,v)$ are l.s.c. for a.e. $t\in (0,T)$;

• 3. $\{v\in B:{\mathfrak {F}}_t(v)\le c\}$ are compact for any $c\ge 0$ and for a.e. $t\in (0,T),$

and a l.s.c. map $g:B\times B\rightarrow [0,+\,\infty ]$ with the property

$$\begin{aligned}&\left[ u,v\in D({\mathfrak {F}}_t),\ g(u,v)=0\right] \Rightarrow u=w,\ \text {for }{\mathrm{a.e.}}\ t\in (0,T). \end{aligned}$$

If

$$\begin{aligned} \sup _{u\in {\mathcal {U}}}\int _0^T{\mathfrak {F}}(t,u(t))\mathrm{d}t<+\,\infty \ \ \text {and}\ \ \lim _{h\downarrow 0}\sup _{u\in {\mathcal {U}}}\int _0^{T-h}g(u(t+h),u(t))\mathrm{d}t=0, \end{aligned}$$

then ${\mathcal {U}}$ is relatively compact in ${\mathscr {M}}(0,T; B).$