Non-equilibrium steady states as saddle points and EDP-convergence for slow-fast gradient systems
- Mielke, Alexander
2020 Mathematics Subject Classification
- 35G25 35K90 47J35 70K70 82C05 92E20
- Slow-fast gradient system, dual and primal dissipation potentials, EDP-convergence, chemical reaction system, reaction-diffusion equation, constrained saddle points, effective kinetic relation
The theory of slow-fast gradient systems leads in a natural way to non-equilibrium steady states, because on the slow time scale the fast subsystem stays in steady states that are driven by the interaction with the slow system. Using the theory of convergence of gradient systems in the sense of the energy-dissipation principle shows that there is a natural characterization of these non-equilibrium steady states as saddle points of a Lagrangian where the slow variables are fixed. We give applications to slow-fast reaction-diffusion systems based on the so-called cosh-type gradient structure for reactions. It is shown that two binary reaction give rise to a ternary reaction with a state-dependent reaction coefficient. Moreover, we show that a reaction-diffusion equation with a thin membrane-like layer convergences to a transmission condition, where the formerly quadratic dissipation potential for diffusion convergences to a cosh-type dissipation potential for the transmission in the membrane limit.