On a nonlocal hyperbolic conservation law arising from a gradient constraint problem

TL;DR

This paper investigates a nonlocal hyperbolic conservation law linked to gradient constraints, establishing existence, uniqueness, and stability of solutions, and providing a framework for solutions to constrained conservation laws.

Contribution

It introduces a novel nonlocal conservation law model for gradient constraints and proves key mathematical properties, advancing understanding of shock prevention mechanisms.

Findings

Existence and uniqueness of solutions established.

Stability of solutions demonstrated via $L^1$-contraction.

A new approach to gradient-constrained conservation laws developed.

Abstract

In some models involving nonlinear conservation laws, physical mechanisms exist which prevent the formation of shocks. This gives rise to conservation laws with a constraint on the gradient of the solution. We approach this problem by studying a related conservation law with a spatial nonlocal term. We prove existence, uniqueness and stability of solution of the Cauchy problem for this nonlocal conservation law. In turn, this allows us to provide a notion of solution to the conservation law with a gradient constraint. The proof of existence is based on a time-stepping technique, and an $L^1$-contraction estimate follows from stability results of Karlsen and Risebro.