Optimal transport with nonlinear mobilities: a deterministic particle approximation result

TL;DR

This paper establishes a rigorous connection between discrete particle approximations and continuous generalized Wasserstein distances with nonlinear mobilities, demonstrating convergence and applications to 1D conservation laws.

Contribution

It provides the first $ ext{Gamma}$-convergence result for discrete metrics approximating generalized Wasserstein distances with nonlinear mobilities on the real line.

Findings

Discrete metrics converge to continuous Wasserstein distances as particle number increases.

The scheme converges at any discrete time step for 1D conservation laws.

New insights into gradient flows and nonlinear mobilities are provided.

Abstract

We study the discretization of generalized Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a $\Gamma$-convergence result for the associated discrete metrics as $N \to \infty$ to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalized minimizing movements, proving a convergence result of the schemes at any given discrete time step $\tau>0$. This the first work of a series aimed at shedding new lights on the interplay between generalized gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities.