Measure solutions, smoothing effect, and deterministic particle approximation for a conservation law with nonlocal flux

TL;DR

This paper develops a new existence and uniqueness theory for a class of nonlocal conservation laws with specific interaction kernels, introduces a dissipative measure solution concept, and proves a deterministic particle approximation with smoothing effects.

Contribution

It introduces the first existence and uniqueness framework for nonlocal conservation laws with a particular kernel and provides a particle approximation method using a novel solution concept.

Findings

Existence and uniqueness of solutions in probability measures.

A dissipative measure solution implies an instantaneous smoothing effect.

A deterministic particle approximation theorem is established.

Abstract

We consider a class of nonlocal conservation laws with an interaction kernel supported on the negative real half-line and featuring a decreasing jump at the origin. We provide, for the first time, an existence and uniqueness theory for said model with initial data in the space of probability measures. Our concept of solution allows to sort a lack of uniqueness problem which we exhibit in a specific example. Our approach uses the so-called \emph{quantile}, or \emph{pseudo-inverse} formulation of the PDE, which has been largely used for similar types of nonlocal transport equations in one-space dimension. Partly related to said approach, we then provide a deterministic particle approximation theorem for the equation under consideration, which works for general initial data in the space of probability measures with compact support. As a crucial step in both results, we use that our concept of solution (which we call \emph{dissipative measure solution}) implies an instantaneous \emph{measure-to-$L^\infty$} smoothing effect, a property which is known to be featured as well by local conservation laws with genuinely nonlinear fluxes.