EDP-convergence for a linear reaction-diffusion system with fast reversible reaction
We study a linear reaction-diffusion system with fast reversible reaction and investigate its behavior if the reaction rate tends to infinity. Importantly, the problem can be understood as a gradient flow in a Wasserstein-type space involving also the reactions in contrast to the classical Wasserstein theory. Using evolutionary Gamma-convergence based on the Energy-Dissipation-Principle (EDP), we show how an effective limit gradient system can be rigorously derived. The limit gradient system induces a reaction-diffusion system with mixed diffusion coefficients coarse-grained with respect to the microscopic equilibrium of the fast reaction. Moreover, we also obtain a gradient structure for the coarse-grained limit system that describes only diffusion with a mixed diffusion coefficient. The result shows a thermodynamic consistent way to derive a coarse-grained model from a multi-scale reaction-diffusion system.