An entropy structure preserving space-time formulation for cross-diffusion systems: Analysis and Galerkin discretization

TL;DR

This paper introduces a novel space-time Galerkin method for cross-diffusion systems that preserves entropy structure, ensuring bounded, nonnegative solutions and providing a rigorous framework for existence, convergence, and numerical approximation.

Contribution

It develops a new space-time formulation and Galerkin discretization that maintain the entropy structure of cross-diffusion systems, with proven existence and convergence results.

Findings

Numerical solutions for porous medium, Fisher-KPP, and Maxwell-Stefan problems.

The method preserves entropy and nonnegativity of solutions.

Convergence of discrete solutions is rigorously established.

Abstract

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.