Nonlocal balance laws – an overview over recent results

Abstract

In this contribution we give a short overview of recent advances in nonlocal conservation laws. In these equations, we assume that the flux f is decomposable into f(x)=V(x)x, x∈R with V being the velocity function. The velocity itself depends nonlocally on the solution, and nonlocality here is realized by an integration over a specific spatial neighborhood. We tackle first problems of existence, uniqueness and regularity of solutions, dwell on the fact that weak solutions are unique (and an entropy condition is not required), and discuss delay nonlocal conservation laws as well as multi-d equations. One key question of nonlocal conservation laws has been whether solutions to these equations converge to the solution of the corresponding local conservation laws when the nonlocality is shrunk to a local evaluation. This has recently been answered positively for a variety of nonlocal impacts, thus closing the gap between nonlocal and local modeling with conservation laws. In contrast to the character of viscosity approximations being parabolic, we keep with the nonlocal approximation a hyperbolic character of the dynamics. This opens a different and novel approach for tackling nonlocal conservation laws.