A new formula for the Wasserstein distance between solutions to (nonlinear) continuity equations

TL;DR

This paper introduces a new formula for the Wasserstein distance between solutions of continuity equations with density-dependent velocities, enabling better quantitative estimates and convergence rates, with applications to porous medium and aggregation-diffusion equations.

Contribution

It provides a novel formula for Wasserstein distance that simplifies analysis of solutions to nonlinear continuity equations and improves convergence rate estimates in various limits.

Findings

Solutions are Lipschitz continuous with respect to the exponent m in the porous medium equation.

Established a convergence rate of 1/√m in the incompressible limit.

Improved the nonlocal-to-local convergence rate from √ε to ε.

Abstract

Given two continuity equations with density-dependent velocities, we provide a new formula for the Wasserstein distance between the solutions in terms of the difference of velocities evaluated at the same density. The formula is particularly attractive to deduce quantitative estimates and rates of convergence for singular limits. We illustrate it using several examples. For the porous medium equation with exponent $m$, we prove that solutions are Lipschitz continuous with respect to $m$, providing a quantitative version of the result of B\'{e}nilan and Crandall. This result can be extended to a general aggregation-diffusion equation. We also study the limit $m \to \infty$ (the so-called mesa problem or the incompressible limit) and we recover the rate of convergence $1/{\sqrt{m}}$. Last but not least, we improve the rate of nonlocal-to-local convergence for the quadratic porous medium equation from recently obtained $\sqrt{\varepsilon}$ to numerically conjectured $\varepsilon$.