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

José A. Carrillo, Piotr Gwiazda, Jakub Skrzeczkowski

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 Bénilan 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 $\varepsilon$.

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

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 , we prove that solutions are Lipschitz continuous with respect to , providing a quantitative version of the result of Bénilan and Crandall. This result can be extended to a general aggregation-diffusion equation. We also study the limit (the so-called mesa problem or the incompressible limit) and we recover the rate of convergence . Last but not least, we improve the rate of nonlocal-to-local convergence for the quadratic porous medium equation from recently obtained to numerically conjectured .

Paper Structure

This paper contains 25 sections, 19 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Motivation.
  3. The main result for 2-Wasserstein gradient flows
  4. The main result for general nonlinear continuity equations.
  5. Applications to the porous medium equation.
  6. Nonlocal-to-local limit
  7. General aggregation-diffusion equation
  8. Structure of the paper
  9. Proofs of Theorem \ref{['thm:simplified_for_2-Wasserstein']} and \ref{['thm:main']}
  10. Proof of Theorem \ref{['thm:simplified_for_2-Wasserstein']}
  11. A Trotter-Kato type estimate
  12. Proof of Theorem \ref{['thm:main']}
  13. Proofs of Theorems \ref{['thm:PME']} and \ref{['thm:PME_incompress_limit']}
  14. Proof of Theorem \ref{['thm:rate_of_conv_nonlocal_to_local']}
  15. Proof of Theorem \ref{['thm:rate_of_conv_nonlocal_to_local']} (the case of $\mathbb{R}$)
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $\Omega$ be $\mathbb{R}^{d}$, its bounded smooth domain or $R\,\mathbb{T}^{d}$. Let $\mu_t, \nu_t \subset \mathcal{P}_2(\Omega)\cap L^1(\Omega)$ be solutions to eq:PDE_continuity_equation_1--eq:PDE_continuity_equation_2 with velocity field $\mathbf{v^{\nu}}[\nu]$ given by eq:particular_vector_fi

Theorems & Definitions (45)

  • Theorem 1.2
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.11
  • Theorem 1.13
  • Remark 1.14
  • Proposition 2.1
  • proof
  • ...and 35 more