Ganesh Vaidya

Aekta Aggarwal, N. K. Aswini, Sarvesh Kumar et al.

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.

Aekta Aggarwal, Ganesh Vaidya

We study the entropy solution for a class of systems of nonlocal conservation laws in which the convective flux is convoluted with a kernel in both spatial and temporal variables. This formulation models the flux dependence on the solution within its spatial neighbourhood (nonlocal in space) as well as on prior states in time (nonlocal in time), thereby incorporating memory effects. In addition, employing a convergent finite volume approximation, the existence of the entropy solution is discussed. The uniqueness of such entropy solutions is also established. In addition, we analyze the asymptotic behaviour of the solutions as the support of the temporal convolution kernel shrinks, demonstrating the "memory-to-memoryless" effect and convergence to the entropy solution of the corresponding nonlocal conservation law without memory (i.e., nonlocal only in space). Convergence rate estimates are derived. In addition, the proposed numerical approximations are shown to be asymptotically compatible with this passage to the memoryless limit by deriving the corresponding asymptotic convergence rate estimates. The analysis is carried out in a very general setting, without imposing any geometric restrictions such as the convexity of the spatial and temporal convolution kernels, unlike the existing literature on the asymptotic analysis of nonlocal-in-space only conservation laws. To the best of our knowledge, this provides the first convergence and asymptotic analysis for finite volume schemes applied to nonlocal conservation laws with memory. Numerical experiments are included to illustrate the theory.