Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

TL;DR

This paper establishes the equivalence between Wasserstein gradient flows and entropy solutions for 1D nonlocal interaction equations, enabling rigorous particle approximations and new insights into the sub-differential structure.

Contribution

It proves the equivalence between gradient flows and entropy solutions in 1D nonlocal PDEs, and introduces a particle system approximation avoiding regularization effects.

Findings

Proves the equivalence between Wasserstein gradient flows and entropy solutions.

Provides a particle-system approximation using wave-front-tracking.

Characterizes the sub-differential of the involved functional.

Abstract

We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow.