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A review of compactness methods for cross-diffusion systems seen as Wasserstein gradient flows

Mathias Dus, Ansgar Jüngel

Abstract

A comprehensive methodology for establishing the existence of gradient flows for cross-diffusion systems with respect to suitable energies is proposed. The approach is based on the construction of piecewise-in-time constant approximations via the Jordan-Kinderlehrer-Otto scheme. Compactness of the approximate sequence is obtained using either the flow interchange technique or the five gradient inequality. These methods are illustrated for both parabolic and hyperbolic-parabolic Busenberg-Travis systems, as well as for several of their variants. This paper reviews the results from the literature and discusses additional properties.

A review of compactness methods for cross-diffusion systems seen as Wasserstein gradient flows

Abstract

A comprehensive methodology for establishing the existence of gradient flows for cross-diffusion systems with respect to suitable energies is proposed. The approach is based on the construction of piecewise-in-time constant approximations via the Jordan-Kinderlehrer-Otto scheme. Compactness of the approximate sequence is obtained using either the flow interchange technique or the five gradient inequality. These methods are illustrated for both parabolic and hyperbolic-parabolic Busenberg-Travis systems, as well as for several of their variants. This paper reviews the results from the literature and discusses additional properties.

Paper Structure

This paper contains 10 sections, 12 theorems, 117 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic elements of optimal transport theory
  3. Parabolic Busenberg--Travis model
  4. Convexity of energies
  5. Existence analysis
  6. A fourth-order Busenberg--Travis system
  7. Hyperbolic--parabolic Busenberg--Travis model
  8. A new definition of solution
  9. One-dimensional equations
  10. Complementary effects from the SKT model

Key Result

Theorem 2

For any couple $(\mu,\nu)\in P_2(\Omega)^2$, there exists a (constant speed) geodesic $(\mu_t)_{0\le t\le 1}\in C^0([0,1];P_2(\Omega))$ connecting $\mu$ and $\nu$. Taking an optimal coupling $\gamma\in\Pi(\mu,\nu)$, a constant speed geodesic is constructed as where $\pi_1$ and $\pi_2$ are the projection operators given by $\pi_1(x,y)=x$, $\pi_2(x,y)=y$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Marginals $u_1(x_1)$, $u_2(x_2)$ (left) and joint density $p(x_1,x_2)$ (right) at time $t=1$.
  • Figure 2: Relative entropy $H(p \,||\, u_1 \otimes u_2)$.

Theorems & Definitions (29)

  • Definition 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 19 more