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A. Jüngel:
"The boundedness-by-entropy method for cross-diffusion systems";
Nonlinearity, 28 (2015), S. 1963 - 2001.

Siehe englisches Abstract.

The global-in-time existence of bounded weak solutions to a large class of
physically relevant, strongly coupled parabolic systems exhibiting a formal
gradient-flow structure is proved. The main feature of these systems is that
the diffusion matrix may be generally neither symmetric nor positive semidefinite.
The key idea is to employ a transformation of variables, determined
by the entropy density, which is defined by the gradient-flow formulation.
The transformation yields at the same time a positive semi-definite diffusion
matrix, suitable gradient estimates as well as lower and/or upper bounds of the
solutions. These bounds are a consequence of the transformation of variables
and are obtained without the use of a maximum principle. Several classes of
cross-diffusion systems are identified which can be solved by this technique.
The systems are formally derived from continuous-time random walks on a
lattice modeling, for instance, the motion of ions, cells, or fluid particles.
The key conditions for this approach are identified and previous results in the
literature are unified and generalized. New existence results are obtained for
the population model with or without volume filling.

Entropy method; cross-diffusion systems

http://dx.doi.org/10.1088/0951-7715/28/6/1963