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

Ansgar Jüngel (1) , Martin Vetter (1) , Antoine Zurek (2)

1 Institute for Analysis and Scientific Computing [Wien]
2 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne

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.

Mots clés: . 2000 Mathematics Subject Classification. 35K20 35K59 35Q92 92D25 Population dynamics cross-diffusion systems nonlocal regularization entropy structure existence of weak solutions uniqueness of solutions large-time behavior of solutions; . 2000 Mathematics Subject Classification. 35K20; 35K59; 35Q92; 92D25 Population dynamics; cross-diffusion systems; nonlocal regularization; entropy structure; existence of weak solutions; uniqueness of solutions; large-time behavior of solutions

Domaines: Equations aux dérivées partielles [math.AP]

Ansgar Jüngel, Martin Vetter, Antoine Zurek. A nonlocal regularization of a generalized Busenberg-Travis cross-diffusion system. 2024. ⟨hal-04630443⟩