交叉扩散系统的一个熵结构保持时空公式：分析和Galerkin离散化
An entropy structure preserving space-time formulation for cross-diffusion systems: Analysis and Galerkin discretization
摘要：
交叉扩散方程组是一类非线性抛物型偏微分方程，在化学浓度和细胞生物学等领域有着广泛的应用。我们提出了一种时空方法来证明交叉扩散系统的有界弱解的存在性，利用系统熵来考察系统的长期行为，并证明该解是非负的，即使当最大值原理不可用时。这种方法自然产生了一种新的时空Galerkin方法的交叉扩散系统的数值逼近，保持其熵结构。我们证明了离散解的存在性和收敛性，并给出了多孔介质，Fisher-KPP和Maxwell-Stefan问题的数值结果。
英文摘要：
Cross-diffusion systems are systems of nonlinear parabolic partial differential equations that are used to describe dynamical processes in several application, including chemical concentrations and cell biology. We present a space-time approach to the proof of existence of bounded weak solutions of cross-diffusion systems, making use of the system entropy to examine long-term behavior and to show that the solution is nonnegative, even when a maximum principle is not available. This approach naturally gives rise to a novel space-time Galerkin method for the numerical approximation of cross-diffusion systems that conserves their entropy structure. We prove existence and convergence of the discrete solutions, and present numerical results for the porous medium, the Fisher-KPP, and the Maxwell-Stefan problem.
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