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

Erstellt aus der Publikationsdatenbank der Technischen Universität Wien.