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A. Jüngel:
"Entropy structure of crossdiffusion models in biology";
Vortrag: Universite de Marseille, Marseille (eingeladen); 31.05.2011.

Siehe englisches Abstract.

Diffusive systems of nonlinear partial differential equations with
cross-diffusion terms appear in many biological fields, like
population dynamics, chemotaxis, and tumor growth. 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, several of 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. This structure is explained for
two examples: a Keller-Segel model for chemotaxis in which
cross-diffusion avoids blow up of the cell density and a tumor-growth
model which describes the encapsulation of tumors. The existence
of global weak solutions is proved and numerical examples are
presented.

Keller-Segel model; tumor-growth model; chemotaxis