EDP-convergence for nonlinear fast-slow reaction systems with detailed balance
- Mielke, Alexander
- Peletier, Mark A.
- Stephan, Artur
2010 Mathematics Subject Classification
- 49S05 47J30 92E20 34E13
- Nonlinear reaction system with detailed balance, fast-reaction limit, gradient structure, gradient system, EDP-convergence, energy-dissipation principle, Gamma-convergence
We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.