Some results on nonlocal conservation laws

Abstract:  In our talk, we will consider nonlocal conservation laws on finite time and space horizon. These classes of conservation laws were introduced in 2006 by [Armbruster, Marthaler, Ringhofer, et al.] and studied for their properties regarding regularity, existence and uniqueness at first by [Coron, Kawski, Wang]. The nonlocal behavior of the introduced models is realized by a flux function depending on the integral of the solution with respect to the space variable. 

Amongst other things the models are used for the simulation and control of the production cycle of semiconductors and, modified, they also play an important role in the simulation of traffic flow. Further applications are imaginable and existent.  

For the named conservation laws we prove existence and uniqueness of solutions in several Banach spaces, and introduce generalizations to problems of multi-commodity flow, of networks, etc.

The proof for the existence of unique weak solutions for these classes of conservation laws - the key part for any further result - strongly relies on the method of characteristics and Banach's fixed-point theorem.