Nonlinear semigroups for nonlocal conservation laws

TL;DR

This paper studies nonlocal conservation laws in multiple dimensions, establishing well-posedness and entropy inequalities using semigroup theory, and demonstrating qualitative properties of solutions.

Contribution

It introduces a framework for analyzing nonlocal conservation laws with broad flux functions, extending classical methods to nonlocal interactions.

Findings

Proved well-posedness for a class of nonlocal conservation laws.

Established that solutions satisfy a Kruzhkov-type entropy inequality.

Demonstrated qualitative properties of solutions similar to local cases.

Abstract

We investigate a class of nonlocal conservation laws in several space dimensions, where the continuum average of weighted nonlocal interactions are considered over a finite horizon. We establish well-posedness for a broad class of flux functions and initial data via semigroup theory in Banach spaces and, in particular, via the celebrated Crandall-Liggett Theorem. We also show that the unique mild solution satisfies a Kru\v{z}kov-type nonlocal entropy inequality. Similarly to the local case, we demonstrate an efficient way of proving various desirable qualitative properties of the unique solution.