Fast-slow chemical reaction systems: gradient systems and EDP-convergenceMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

In my talk, we start with introducing gradient systems and EDP-convergence. A gradient system (X, E, R) consists of a state space Q ⊂ X (X is a separable, reflexive Banach space), an energy functional E : Q → R ∪ {∞} and a dissipation potential R : Q × X → [0, ∞[, which is convex, lower semicontinuous and satisfies R(·, 0) = 0. The associated gradient-flow equation is then given by 0 ∈ ∂R(u(t), u′(t)) + DE(u(t)) or equivalently u′(t) ∈ ∂R∗(u(t), −DE(u(t))). The energy-dissipation principle (EDP) then states, that the gradient-flow evolution equation is equivalent to the energy-dissipation balance of the form E(u(t)) + D(u) = E(u(0)), D(u) = ∫ T 0 R(u, ̇u) + R∗(u, −DE(u)) dt. For families of gradient systems (Q, E, R), a structural convergence, the so-called ”EDP-convergence” has been developed in recent years. It provides many interesting features from the mathematical as well as the modeling perspective. As an application of the general theory, we consider reaction and reaction-diffusion systems satisfying mass-action kinetics in the singular limit of slow and fast reactions. Instead of investigating solely the evolution equations, our analysis uses methods from the calculus of variations and relies on the energy-dissipation principle of the associated gradient systems. We show that an effective gradient structure can be rigorously derived via EDP-convergence.