Persisting entropy structure for nonlocal cross-diffusion systems
Résumé
For cross-diffusion systems possessing an entropy (i.e. a Lyapunov functional)
we study nonlocal versions and exhibit sufficient conditions to ensure that the
nonlocal version inherits the entropy structure. These nonlocal systems can be
understood as population models per se or as approximation of the classical ones.
With the preserved entropy, we can rigorously link the approximating nonlocal
version to the classical local system. From a modelling perspective this gives a
way to prove a derivation of the model and from a PDE perspective this provides
a regularisation scheme to prove the existence of solutions. A guiding example is
the SKT model [22]. In this context we answer positively the question raised by
Fontbona and Méléard [12] for the derivation and thus complete the derivation.
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