Stability estimates for systems of nonlocal balance laws with memory
Provides theoretical stability guarantees for a broad class of PDEs with memory, relevant for mathematicians working on nonlocal conservation laws.
The authors establish stability estimates for entropy solutions of systems of nonlocal balance laws with memory, showing dependence on perturbations in flux, kernels, and initial data, with numerical experiments confirming the theory.
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.