Coarse‐graining via EDP‐convergence for linear fast‐slow reaction‐diffusion systems

In [7] a fast‐reaction limit for a linear reaction‐diffusion system consisting of two diffusion equations coupled by a linear reaction is performed. The linear reaction‐diffusion system is understood 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 system 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 scalar diffusion equation with a mixed diffusion constant for the coarse‐grained slow variable.

complemented with no-flux boundary conditions and initial conditions, where δ 1 , δ 2 > 0 are diffusion coefficients for species X 1 and X 2 , respectively, and α, β > 0 are reaction rates describing the reaction speed of the linear reaction X 1 ⇋ X 2 . We are interested in the limit ε → 0, i.e. if the reaction is much faster than the diffusion Reaction systems and reaction-diffusion systems with slow and fast time scales have attracted a lot of attention in the last decades. In [1] the following fast-reaction limit for ε → 0 is proved: Let c ε 1 and c ε 2 be weak solutions of (LRDS). Then c ε 1 → c 1 and c ε 2 → c 2 in L 2 ([0, T ] × Ω) as ε → 0, and we have c1 β = c2 α . Moreover, defining the coarse-grained concentration c = c 1 + c 2 , thenĉ solves the diffusion equationċ =δ∆ĉ with a new mixed diffusion coefficientδ = βδ1+αδ2 α+β .

Gradient structure for linear reaction-diffusion systems
In [7], we were not primarily interested in convergence of the solutions of (LRDS). Instead, we perform the fast-reaction limit on the level of the underlying variational structure, which then implies convergence of solutions as a byproduct. The idea is that reaction-diffusion systems such as (LRDS) can be written as a gradient flow equation induced by a gradient system • the driving functional is the free energy E(µ) = Ω 2 j=1 E B cj wj w j dx for measures µ = c dx, with the Boltzmann function E B (r) = r log r − r + 1 and the (in general space dependent) stationary measure w = (w 1 , w 2 ) T .
• the dissipation potential R * ε determining the geometry of the underlying space is given by two parts R * ε = R * diff +R * react,ε describing the diffusion and reaction separately.
induces the Wasserstein distance on Q following the pioneering work of Otto and coauthors [3]. Later Mielke [4] proposed a quadratic gradient structure with the same driving functional also for reaction-diffusion systems with reversible reactions satisfying detailed balance. A different but related gradient structure is the the so-called cosh-type gradient structure, where the reaction part is given by react,ε , the reaction-diffusion system (LRDS) can now be formally written as a gradient flow equationμ = ∂ ξ R * ε (µ, −DE(µ)).

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Section 14: Applied analysis

EDP-convergence with tilting
In [7] an effective gradient system (Q, E, R * eff ) is constructed such that (Q, E, R * ε ) → (Q, E, R * eff ) as ε → 0. For this, the notion of EDP-convergence as introduce in [2] was used. It is based on the dissipation functional D η ε (µ) = T 0 R ε (µ,μ) + R * ε (µ, η − DE(µ)) dt, which, for solutions µ of the gradient flow equation describes the total dissipation between initial time E(µ(0)) and final time E(µ(T )), and can now be defined for general trajectories µ ∈ L 1 ([0, T ], Q). Here, the primal dissipation potential R ε is defined as the Legendre transform of R * ε with respect to the second variable. The notion of EDPconvergence with tilting requires Γ-convergences of the energies E ε Γ − → E 0 and of the dissipation functionals D η ε Γ − → D η 0 in suitable topologies, such that for all tilts η the limit D η 0 has the form D η 0 (µ) = T 0 R eff (µ,μ) + R * eff (µ, η − DE 0 (µ)) dt. Importantly, the effective dissipation potential R eff in the Γ-limit is independent of the tilts, which in our situation correspond to an external potential V = (V 1 , V 2 ) added to the energy E, i.e. E V := E + V On the level of the PDE, the original reaction-diffusion system (LRDS) is extended to a reaction-drift-diffusion system of the form d dt

Effective gradient system and coarse-grained gradient system
The main result of [7] asserts tilt EDP-convergence of (Q, E, R * ε ) to (Q, E, R * eff ) as ε → 0 and the effective dissipation potential is given by R * eff = R * diff + χ {ξ1=ξ2} (where χ A is the characteristic function of convex analysis). The effective dissipation potential describes diffusion but restricts the chemical potential ξ = −DE V (µ) to a linear submanifold given by is given as the minimum of E V on Q. The induced gradient flow equation of the gradient system (Q, E, R * eff ) is then given by a system of drift-diffusion equations on a linear submanifold with a space and time dependent Lagrange multiplier λ Similarly to the space-independent situation of linear [6] or nonlinear [5] fast-slow reaction systems, the effective gradient system can be equivalently described in terms of coarse-grained slow variables. The coarse-grained gradient system is given by (Q,Ê,R * ), whereQ = Prob(Ω) is the coarse-grained state space, the driving functional and the dissipation functional are given byÊ with the reconstructed measure (µ 1 , µ 2 ) = ( and the mixed potentialV = − log( β α+β e −V1 + α α+β e −V2 ). The gradient flow equation induced by (Q,Ê,R * ) is given bẏ c = div δV ∇ĉ +δ Vĉ ∇V , which is in accordance with [1] in the potential-free case.
Summarizing, the results from [7] show how to obtain a coarse-grained model for a multi-scale problem by performing structural convergence on the level of the gradient system.