EDP-convergence for a linear reaction-diffusion system with fast reversible reaction

TL;DR

This paper analyzes the fast-reaction limit of a linear reaction-diffusion system using gradient flow structures, demonstrating EDP-convergence and deriving a coarse-grained diffusion system with Lagrange multipliers.

Contribution

It introduces a novel approach to analyze the fast-reaction limit via EDP-convergence within a gradient flow framework, including the derivation of a coarse-grained diffusion system.

Findings

EDP-convergence established for the reaction-diffusion system

Limit system characterized by a diffusion equation with Lagrange multipliers

Coarse-grained system described by a diffusion equation with mixed diffusion constant

Abstract

We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measure equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable.