The boundedness-by-entropy principle for cross-diffusion systems

By A. Jungel. 2014

50 citations recorded by the source corpus.

A novel principle is presented which allows for the proof of bounded weak solutions to a class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea of the principle 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 semidefinite 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.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/74bb6725a09b7c1fb6a20955eaa784c8f2a00ff1

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In Applied Mathematics

In Modeling and Simulation

In Public Health, Environmental and Occupational Health

Citations in the corpus

On the Entropic Structure of Reaction-Cross Diffusion Systems (2014, methodology): “In our framework, we expect positive solutions, that is D = R+ × R ∗ +, but it is sometimes useful to consider bounded sets for D, this is for instance the case in [11].”