Existence and uniqueness of nonlocal nonlinear conservation laws via fixed-point methods

TL;DR

This paper establishes the existence and uniqueness of solutions for nonlocal nonlinear conservation laws using fixed-point methods, extending previous models and including numerical simulations.

Contribution

It introduces a fixed-point framework for scalar conservation laws with nonlocal flux dependence, covering models with memory effects and delays.

Findings

Proved local-in-time existence and uniqueness of entropy solutions.

Extended the framework to global-in-time solutions under additional assumptions.

Numerical simulations demonstrate the impact of memory effects on solution behavior.

Abstract

We investigate the well-posedness of scalar conservation laws whose flux depends on the solution both pointwise and nonlocally through integral averages. Our analysis is based on a fixed-point formulation, in which the nonlocal dependence is incorporated as a space- and time-dependent component of the flux, together with classical stability estimates for entropy solutions. This framework unifies and extends several models previously considered in the literature and applies, in particular, to conservation laws with memory effects (nonlocality in time) or delay. We prove the existence and uniqueness of weak entropy solutions on a sufficiently short time horizon and show that under additional assumptions, existence and uniqueness can be obtained on any finite time horizon. In addition, we present numerical simulations to illustrate the qualitative effects of memory on the solution dynamics.