Well-posedness of multidimensional nonlocal conservation laws with nonlinear mobility and bounded force

TL;DR

This paper proves local existence and uniqueness of solutions for multidimensional nonlocal conservation laws with nonlinear mobility, showing solutions can develop shocks and providing a convergence rate for the vanishing viscosity approximation.

Contribution

It extends well-posedness results to nonlinear mobility cases in multiple dimensions under weak kernel assumptions, including shock formation analysis.

Findings

Solutions may develop shocks in finite time

Established local-in-time existence and uniqueness

Provided convergence rate for vanishing viscosity approximation

Abstract

We establish local-in-time existence and uniqueness results for nonlocal conservation laws with a nonlinear mobility, in several space dimensions, under weak assumptions on the kernel, which is assumed to be bounded and of finite total variation. Contrary to the linear mobility case, solutions may develop shocks in finite time, even when the kernel is smooth. We construct entropy solutions via a vanishing viscosity method, and provide a rate of convergence for this approximation scheme.