Nonlocal Conservation Laws
Vortragender: Alexander Keimer
We will discuss recent advances on nonlocal conservation laws, where part of the
flux function does not depend point-wise on the density but on an integration of
the density around the given space point. This class of equations will be
illustrated and motivated by applications in traffic flow modelling, pedestrian
flow dynamics and more. With a fixed-point approach in the proper Banach space
we establish existence and uniqueness of weak solutions under rather general
assumptions and render the often used Entropy condition in this context
obsolete. Nonlocal conservation laws with time delay as well as discontinuous
(in space) nonlocal conservation laws are also covered by the established theory
and will be investigated.
Finally, the singular limit problem, i.e., the question whether the solution of
the nonlocal conservation law converges to the Entropy solution of the (local)
conservation law is studied.
Indeed, for specific kernels such a result holds which closes the gap between
local and nonlocal conservation laws.