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A nonlocal regularization of a generalized Busenberg-Travis cross-diffusion system
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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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Nonlocal approximation of an anisotropic cross-diffusion system
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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