Well posedness of general cross-diffusion systems
Résumé
The paper is devoted to the mathematical analysis of the Cauchy problem for general cross-
diffusion systems without any assumption about its entropic structure. A global existence result of nonneg-
ative solutions is obtained by applying a classical Schauder fixed point theorem. The proof is upgraded for
enhancing the regularity of the solution, namely its gradient belongs to the space Lr((0, T ) × Ω) for some
r > 2. To this aim, the Schauder’s strategy is coupled with an extension of Meyers regularity result for
linear parabolic equations. We show how this approach allows to prove the well-posedness of the problem
using only assumptions prescribing and admissibility range for the ratios between the diffusion and cross-
diffusion coefficients. The results are compared with those that are reachable with an additional regularity
assumption on the parabolic operator, namely a small BMO assumption for its coefficients. Finally, the
question of the maximal principle is also addressed, especially when source terms are incorporated in the
equation in order to ensure the confinement of the solution
Origine | Fichiers produits par l'(les) auteur(s) |
---|