EDP-convergence for a linear reaction-diffusion system with fast reversible reaction
- Stephan, Artur
2010 Mathematics Subject Classification
- 49S05 47J30 35A15 5K57 92E20 35Q84
- Markov process with detailed balance, linear reaction-diffusion system, gradient systems, gradient flows, evolutionary convergence, Energy-Dissipation-Balance, coarse-graining, microscopic equilibrium, Gamma-convergence
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 measures 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.