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A. Jüngel:
"Entropy structure in cross-diffusion models from biology";
Vortrag: Vortragsreihe, University of Dundee (eingeladen); 20.01.2014.

Abstract:
Multi-particle systems for multiple species or fluid components
can be described in the continuum limit by cross-diffusion systems,
derived from lattice or fluid-type models. The main feature of these
strongly coupled partial differential equations is that the diffusion
matrix is often neither symmetric nor positive definite, which makes
the mathematical analysis very challenging.

In many situations, however, these systems possess an entropy
structure, i.e., there exist so-called entropy variables which make
the diffusion matrix positive definite. The existence of these
variables is equivalent to the existence of a Lyapunov functional
(free energy or logarithmic entropy), leading to a priori estimates.
Although the maximum principle does not hold for such systems, the
entropy variables naturally leads to lower or upper bounds of the
solution. This allows for a mathematical theory for certain classes
of cross-diffusion systems.

We detail this theory for some examples: a variant of the
chemotaxis Keller-Segel model; the tumor-growth model by Jackson
and Byrne; and Maxwell-Stefan systems for multicomponent mixtures.
The existence of global weak solutions is proved and numerical
examples are presented.