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A nonlocal regularization of a generalized Busenberg-Travis cross-diffusion system

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arxiv 2407.01123 v1 pith:LLPSPURG submitted 2024-07-01 math.AP

classification math.AP
keywords solutionssystemregularizationbusenberg-traviscross-diffusiongeneralizednonlocaluniqueness
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A cross-diffusion system with Lotka-Volterra reaction terms in a bounded domain with no-flux boundary conditions is analyzed. The system is a nonlocal regularization of a generalized Busenberg-Travis model, which describes segregating population species with local averaging. The partial velocities are the solutions of an elliptic regularization of Darcy's law, which can be interpreted as a Brinkman's law. The following results are proved: the existence of global weak solutions; localization limit; boundedness and uniqueness of weak solutions (in one space dimension); exponential decay of the solutions. Moreover, the weak-strong uniqueness property for the limiting system is shown.

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  1. Nonlocal approximation of an anisotropic cross-diffusion system

    math.AP 2024-12 conditional novelty 6.0 of 10

    Weak solutions of an anisotropic nonlocal cross-diffusion system converge to weak solutions of the corresponding local cross-diffusion system in the vanishing viscosity limit.

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