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

TL;DR

The paper addresses the well-posedness of multidimensional nonlocal conservation laws with nonlinear mobility and bounded forcing, focusing on existence, uniqueness, and stability of entropy solutions. It employs a vanishing-viscosity approach to construct entropy solutions and proves uniform BV and L1 estimates, yielding global well-posedness under minimal kernel regularity. A key contribution is a 1/2-order convergence rate for viscous approximations and the propagation of BV when kernels are in BV, including certain singular kernels like Riesz-type with s<d−2. The results extend to one-dimensional Keller–Segel and related nonlocal models, establishing uniqueness of entropy solutions in those classical settings and providing a rigorous framework for nonlocal transport with nonlinear mobility.

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.

Theorems & Definitions (35)

• Definition 1.2: Entropy solutions
• Remark 1.3
• Definition 1.4: Vanishing viscosity solutions
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10
• Proposition 2.1: Local well-posedness of \ref{['eq:viscousPDE']}