Measure solutions, smoothing effect, and deterministic particle approximation for a conservation law with non-local flux

Abstract

We consider a class of non-local 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 us to sort a lack of uniqueness problem which we exhibit in a specific example. Our approach uses the so-called quantile, or pseudo-inverse, formulation of the PDE, which has been largely used for similar types of non-local 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 dissipative measure solution) implies an instantaneous measure-to-$L^{\infty }$ smoothing effect, a property which is known also to be featured by local conservation laws with genuinely non-linear fluxes.