Well posedness of general cross-diffusion systems
| Autor: | Carole Rosier, Catherine Choquet, Lionel Rosier |
|---|---|
| Přispěvatelé: | Mathématiques, Image et Applications - EA 3165 (MIA), Université de La Rochelle (ULR) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Applied Mathematics
Operator (physics) 010102 general mathematics Structure (category theory) Space (mathematics) 01 natural sciences Parabolic partial differential equation 010101 applied mathematics Range (mathematics) Schauder fixed point theorem Initial value problem Applied mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] 0101 mathematics Diffusion (business) Analysis ComputingMilieux_MISCELLANEOUS Mathematics |
| Zdroj: | Journal of Differential Equations Journal of Differential Equations, Elsevier, 2021, 300, pp.386-425. ⟨10.1016/j.jde.2021.08.001⟩ |
| ISSN: | 0022-0396 1090-2732 |
| Popis: | 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 nonnegative 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 L r ( ( 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. |
| Databáze: | OpenAIRE |
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