Abstract
In this note, necessary and sufficient conditions for the existence of an entropy structure for certain classes of cross-diffusion systems with diffusion matrix A(u) are given, 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 constant up to nonlinear perturbations, the existence of an entropy structure is equivalent to the normal ellipticity of A. The results are applied to various examples from physics and biology. Finally, the normal ellipticity of the n-species population model of Shigesada, Kawasaki, and Teramoto is proved.
Highlights
Cross-diffusion systems are systems of quasilinear parabolic equations in which the gradient of one variable induces a flux of another variable
It turns out that there might exist a transformation of variables such that the transformed diffusion matrix becomes positive definite and sometimes even symmetric
The question is under which conditions does such a transformation exist? In this paper, we will give some necessary and sufficient conditions for the existence of entropy variables for certain classes of crossdiffusion systems
Summary
Cross-diffusion systems are systems of quasilinear parabolic equations in which the gradient of one variable induces a flux of another variable. Section 4: If the diffusion matrix A is constant, its normal ellipticity is equivalent to the existence of an entropy structure. Section 5: If the entropy density is the sum of single-valued functions and h′′(u)A(u) is symmetric, the positive definiteness of h′′(u)A(u) is equivalent to the positivity of the leading principal minors of A(u) This avoids the computation of the eigenvalues of A(u) to check its normal ellipticity. There exists a second entropy density, which is of quadratic type It is derived from the Poincare lemma for closed differential forms by interpreting the detailed-balance condition as the curl-freeness of the vector-field We investigate the normal ellipticity of the diffusion matrix (5) of the volume-filling model and (6) of the fluid mixture model
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