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A. Jüngel:
"Entropy structure of cross-diffusion systems in biology";
Hauptvortrag: First Graz-Wien Bio-PDE Day, Graz (eingeladen); 08.05.2012.

Siehe englisches Abstract.

Diffusive systems of nonlinear partial differential equations with
cross-diffusion terms appear in many biological and physical
applications like chemotaxis, tumor modeling, and gas dynamics.
The main feature of these systems is that the diffusion matrix
is generally neither symmetric nor positive definite, which
makes the mathematical analysis very challenging. Interestingly,
these systems possess an entropy structure, i.e., there exist
so-called entropy variables which make the diffusion matrix symmetric
and positive definite. The existence of these variables is equivalent
to the existence of an entropy which turns out to be a Lyapunov
functional and leads to a priori estimates. Although the maximum
principle does not hold for such systems, the entropy variables
naturally lead to lower or upper bounds of the solution.
This structure is explained for three examples: a Keller-Segel model
for chemotaxis with additional cross-diffusion; a tumor-growth model
which describes tumor encapsulation; and the Maxwell-Stefan system
for multicomponent gas mixtures. The existence of global weak
solutions is proved and numerical examples are presented.

Keller-Segel model; tumor growth model