On multidimensional nonlocal conservation laws with BV kernels

TL;DR

This paper studies nonlocal conservation laws with BV kernels, proving local existence and uniqueness, and showing that solutions can blow up in finite time, with different approximations leading to different measures after blow-up.

Contribution

It establishes existence and uniqueness results for nonlocal conservation laws with BV kernels and demonstrates finite-time blow-up phenomena.

Findings

Solutions can blow up in finite time.

Different smooth kernel approximations lead to different measures after blow-up.

Existence and uniqueness are proved under weak differentiability assumptions.

Abstract

We establish local-in-time existence and uniqueness results for nonlocal conservation laws in several space dimensions under weak (that is, Sobolev or BV) differentiability assumptions on the convolution kernel. In contrast to the case of a smooth kernel, in general the solution experiences finite-time blow-up. We provide an explicit example showing that solutions corresponding to different smooth approximations of the convolution kernel in general converge to different measures after the blow-up time. This rules out a fairly natural strategy for extending the notion of solution of the nonlocal conservation law after the blow-up time.