Existence, uniqueness and regularity results on nonlocal balance laws

Abstract

We consider a class of nonlocal balance laws as initial value problems on a finite time horizon and show existence and uniqueness of the corresponding weak solutions. The description “nonlocal” refers to the velocity of the balance law that depends on the weighted integral over an area in space at any given time. Existence of a weak solution for initial data and right hand side data in L1∩L∞, in L∞ and in special cases in L1 is shown via the method of characteristics, resulting in a fixed-point problem in the nonlocal term. The uniqueness of a weak solution with relatively weak assumptions on the flux function and the nonlocal term is established, so that the uniqueness result does not require the well-known “Kružkov” entropy condition as it is typical for (local) balance laws and was up to now used in the available literature also for nonlocal balance laws.