Stability estimates for systems of nonlocal balance laws with memory

TL;DR

This paper develops stability estimates for entropy solutions of nonlocal balance law systems that incorporate spatial and temporal memory effects, supported by numerical experiments.

Contribution

It provides the first stability estimates for systems with combined spatial and temporal nonlocality involving convolution-based fluxes and sources.

Findings

Stability estimates quantify solution sensitivity to kernel and initial data perturbations.

Memory effects significantly influence solution dynamics as shown in numerical experiments.

Abstract

In this work, we investigate entropy solutions for a class of systems of nonlocal {balance laws in which the convective flux and the source involves terms where the state variable convolved with kernels} in both spatial and temporal variables. This formulation captures the dependence of the flux on the solution within its spatial neighborhood (spatial nonlocality) as well as on its past states (temporal nonlocality), thereby incorporating memory effects. The resulting systems are coupled through these nonlocal interactions. We establish stability estimates for entropy solutions with respect to perturbations in the flux, the spatial and temporal kernels, and the initial data for the corresponding initial value problems. Finally, we present numerical experiments to illustrate the theoretical results and to highlight the influence of memory and source terms on the solution dynamics.