论文标题
针对具有熵结构的一类交叉扩散系统有限解决方案的部分Hölder规律性
Partial Hölder Regularity for Bounded Solutions of a Class of Cross-Diffusion Systems with Entropy Structure
论文作者
论文摘要
在此贡献中，我们获得了部分$ c^{0，α} $ - 针对特定类别的交叉扩散系统有限解决方案的规律性，这些解决方案是强烈耦合的，退化的准抛物线系统。在稍微限制的假设下，我们获得了部分$ c^{1，α} $ - 规律性。我们认为的交叉扩散系统具有正式的梯度流结构，从某种意义上说，它们与凸熵功能的梯度流正式相同。此外，我们假设交叉扩散系统不是填充体积。我们在此贡献中介绍的主要新颖工具是“胶合熵密度”，它使我们能够在这种新环境中模仿Giaquinta和Struwe的非线性抛物线系统的部分Hölder规律性的经典理论。为了证明我们的结果的适用性，我们给出了满足我们假设的两个跨扩散系统的两个示例 - 其中一个是人口动力学的两个组成部分Shigesada-Kawasaki-teramoto（SKT）模型。
In this contribution we obtain partial $C^{0,α}$-regularity for bounded solutions of a certain class of cross-diffusion systems, which are strongly coupled, degenerate quasilinear parabolic systems. Under slightly more restrictive assumptions, we obtain partial $C^{1,α}$-regularity. The cross-diffusion systems that we consider have a formal gradient flow structure, in the sense that they are formally identical to the gradient flow of a convex entropy functional. Furthermore, we assume that the cross-diffusion systems are not volume-filling. The main novel tool that we introduce in this contribution is a "glued entropy density," which allows us to emulate the classical theory of partial Hölder regularity for nonlinear parabolic systems by Giaquinta and Struwe within this new setting. To demonstrate the applicability of our results, we give two examples of well-studied cross-diffusion systems that satisfy our assumptions --one of which is the two component Shigesada-Kawasaki-Teramoto (SKT) model for population dynamics.
