2020/07/07 by Marcel Braukhoff, Braukhoff, Marcel, Claudia Raithel +3
Mathematics · Medicine · #35B65 #35K67 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2007.03561
openalex publication_date 2020/07/07 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
In this contribution we obtain partial C0,\α-regularity for bounded\nsolutions of a certain class of cross-diffusion systems, which are strongly\ncoupled, degenerate quasilinear parabolic systems. Under slightly more\nrestrictive assumptions, we obtain partial C1,\α-regularity. The\ncross-diffusion systems that we consider have a formal gradient flow structure,\nin the sense that they are formally identical to the gradient flow of a convex\nentropy functional. Furthermore, we assume that the cross-diffusion systems are\nnot volume-filling. The main novel tool that we introduce in this contribution\nis a "glued entropy density," which allows us to emulate the classical theory\nof partial H "older regularity for nonlinear parabolic systems by Giaquinta\nand Struwe within this new setting. To demonstrate the applicability of our\nresults, we give two examples of well-studied cross-diffusion systems that\nsatisfy our assumptions --one of which is the two component\nShigesada-Kawasaki-Teramoto (SKT) model for population dynamics.\n