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X. Chen, A. Jüngel:
"When do cross-diffusion systems have an entropy structure ?";
in: "ASC Report 22/2019", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2019, ISBN: 978-3-902627-12-4, 1 - 23.

Necessary and sufficient conditions for the existence of an entropy structure for certain classes of cross-diffusion systems with diffusion matrix A(u) are derived, based on results from matrix factorization. The entropy structure is important in the analysis
for such equations since A(u) is typically neither symmetric nor positive definite. In particular, the normal ellipticity of A(u) for all u and the symmetry of the Onsager matrix implies its positive definiteness and hence an entropy structure. If A is constant
or nearly constant in a certain sense, the existence of an entropy structure is equivalent to the normal ellipticity of A. Several applications and examples are presented, including the n-species population model of Shigesada, Kawasaki, and Teramoto, a volume-filling model, and a fluid mixture model with partial pressure gradients. Furthermore, the normal elipticity of these models is investigated and some extensions are discussed.

Cross diffusion, entropy method, normal ellipticity, matrix factorization, Lyapunov equation, population model, volume-filling model, fluid mixture model.

http://www.asc.tuwien.ac.at/preprint/2019/asc22x2019.pdf