Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping

Abstract

In this article, we generalize some existence and uniqueness results in [22] for scalar nonlocal balance laws to multi-dimensional nonlocal balance laws. Also in the multi-dimensional case it is possible to get rid of the Entropy condition which is usually postulated to guarantee a unique weak solution of (local) balance laws but which we prove to be redundant in the nonlocal case. We start with a well known existence theorem for linear balance laws and the explicit formula for a solution to obtain a fixed-point problem in the nonlocal term. Using Banach's fixed-point theorem we prove existence and uniqueness of a fixed-point. This existence and uniqueness result is used for constructing and proving a unique solution of the multi-dimensional nonlocal balance law on a sufficiently small time horizon. While in many publications the nonlocal term will be considered over the entire space domain, we present an existence and uniqueness result which allows rather general integration areae for the nonlocal term. Eventually, we obtain a unique weak solution on every finite time horizon. Existence and uniqueness are complemented by stability results and an exhaustive study on more regular weak solutions in case the data is more regular.