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Entropic structure and duality for multiple species cross-diffusion systems. (English) Zbl 1366.35065
Summary: This paper deals with the existence of global weak solutions for a wide class of (multiple species) cross-diffusions systems. The existence is based on two different ingredients: an entropy estimate giving some gradient control and a duality estimate that gives naturally \(L^2\) control. The heart of our proof is a semi-implicit scheme tailored for cross-diffusion systems firstly defined in [L. Desvillettes et al., Commun. Partial Differ. Equations 40, No. 9, 1705–1747 (2015; Zbl 1331.35179)] and a (nonlinear Aubin-Lions type) compactness result developed in [the second author, J. Evol. Equ. 16, No. 1, 65–93 (2016; Zbl 1376.46015)] and [B. Andreianov, C. Cancès and the second author, “A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs”, Preprint, arXiv:1504.03891] that turns the (potentially weak) gradient estimates into almost everywhere convergence. We apply our results to models having an entropy relying on the detailed balance condition exhibited by X. Chen, E. S. Daus and A. Jüngel [“Global existence analysis of cross-diffusion population systems for multiple species”, Preprint, arXiv:1608.03696].

MSC:
35K40 Second-order parabolic systems
35K57 Reaction-diffusion equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B60 Continuation and prolongation of solutions to PDEs
35A35 Theoretical approximation in context of PDEs
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