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

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.