EDP-convergence for a linear reaction-diffusion system with fast reversible reaction

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 measure 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.

In this work, we are not primary interested in convergence of solutions of system (1.2). Instead, we perform the fast-reaction limit on the level of the underlying variational structure, which then implies convergence of solutions as a byproduct. Our starting point is that reaction-diffusion systems such as (1.2) can be written as a gradient flow equation induced by a gradient system (Q, E, R * ε ), where the state space Q is the space of probability measures Q = Prob(Ω × {1, 2}) and the driving functional is the free-energy E(µ) = Ω 2 j=1 E B c j w j 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 that determines the geometry of the underlying space is given by two parts R * ε = R * diff + R * react,ε describing the diffusion and reaction separately. Since the pioneering work of Otto and coauthors [JKO98,Ott01] it is known that diffusion has to be understood as a gradient system driven by the free-energy in the space of probability measures equipped with the Wasserstein distance. The corresponding dissipation potential R * diff is quadratic and given by 1 ε Ω C * (ξ 1 (x) − ξ 2 (x)) dµ 1 dµ 2 , with C * (r) = 4(cosh(r/2) − 1). Setting R * ε = R * diff + R * react,ε , the reaction-diffusion system (1.2) can now be written as a gradient flow equatioṅ µ = ∂ ξ R * ε (µ, −DE(µ)).
Although there are many gradient structures for (1.2) (see e.g. [MiS20,Sect. 4]) and the cosh-type gradient structure entails several technical difficulties as defining a nonlinear kinetic relation and not inducing metric on Q, it nevertheless has several significant features. Historically, it has its origin in [Mar15] where, following thermodynamic considerations, chemical reactions are written in exponential terms. In recent years, the cosh-gradient structure has been derived via a large-deviation principle [MPR14, MP * 17], and it was shown that it is stable under limit processes [LM * 17] that are similar to our approach. Moreover, it does not explicitly depend on the stationary measure w, which allows for an rigorous distinction between the energetic and dissipative part [MiS20]. This is physically reasonable because a change of the energy by an external field should not influence the geometric structure of the underlying space. The goal of the paper is to construct an effective gradient system (Q, E, R * eff ) and perform the limit (Q, E, R * ε ) → (Q, E, R * eff ) as ε → 0. For this, we use the notion of convergence of gradient systems in the sense of the energy-dissipation principle, shortly called EDP-convergence. EDP-convergence was introduced in [DFM18] and further developed in [MMP20,MiS20] and is based on the dissipation functional 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, see Section 2.2 for a precise definition. Importantly, the effective dissipation potential R eff in the Γ-limit is independent of the tilts, hence allowing for extended energies. In our situation, the tilts η correspond to an external potential V = (V 1 , V 2 ) added to the energy E. On the level of the PDE, the starting reaction-diffusion system is extended to a reaction-drift-diffusion system of the form d dt The main result of the paper is Theorem 4.3 which asserts tilt EDP-convergence of (Q, E, R * ε ) to (Q, E, R * eff ) as ε → 0 where the effective dissipation potential is given by where χ A is the characteristic function of convex analysis taking values zero in A and infinity otherwise. The effective dissipation potential describes diffusion but restricts the chemical potential ξ = (ξ 1 , ξ 2 ) to a linear submanifold. 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 Moreover, as an immediate consequence of Theorem 4.3, we obtain that the effective gradient flow equation can be equivalently described as a drift-diffusion equation of the coarse-grained concentrationĉ, see Proposition 4.5. Introducing the mixed diffusion co- which is in accordance with [BoH02] in the potential-free case V = const. Moreover, we obtain a natural coarse-grained gradient structure (Q,Ê,R * ), whereQ = Prob(Ω) is the coarse-grained state space andÊ,R * are the coarse-grained energy functional and dissipation potential, respectively. Interestingly, this coarse-grained gradient structure (Q,Ê,R * ) contains the same information as the effective gradient structure (Q, E, R * eff ), although defined on a smaller state space, see Proposition 4.5.
The result on tilt EDP-convergence is an immediate consequence of the Γ-convergence result of the dissipation functional D η ε (Theorem 5.12). The primal dissipation potential R ε is given by an infimal sum consisting of diffusion fluxes and reaction fluxes coupled via a generalized continuity equation, see Section 3.3. Theorem 5.12 follows from the following observations: R * ε converges monotonically to a singular limit R * eff , the primal dissipation potentials R ε degenerate. It is not possible to control the rates ofμ 1 andμ 2 separately by R ε , since the reaction flux between both species may become unbounded. Instead, it is possible to prove compactness for the sum (or slow variable) µ 1 + µ 2 by R ε , and proving convergence towards the slow-manifold where an equilibration takes place, i.e. αc ε 1 − βc ε 2 → 0. The two pieces of complementary information provide strong convergence of the densities c ε in L 1 ([0, T ] × Ω). This procedure has been already successfully applied for linear and nonlinear reaction systems [MiS20,MPS20] and is here applied to a space-dependent evolution system. A posteriori we conclude that the limit measure µ 0 has indeed an absolutely continuous representative using results from [AGS05]. The construction of the recovery sequence relies on the fact that the limit dissipation functional can be equivalently written as a functional of coarse-grained variables. Only the reaction flux, which is present for positive ε > 0 and hidden for ε = 0, has to be reconstructed. One observes that diffusion causes the reaction flux on an infinitesimally small scale. Since the dissipation functional considers also fluctuations which may be not strictly positive and not smooth in contrast to the solution of the linear reaction diffusion system (1.2), the construction of a recovery sequence is completed by a suitable approximation argument.
Let us finally mention, that the same results can also be established for reactiondiffusion systems, where more than two species are involved. Applying the coarse-graining and reconstruction machinery as developed in [MiS20], a similar Γ-convergence result for the dissipation functional can be proved. For notational convenience we restrict to the two-species situation and briefly discuss the multi-species case in Section 6. We refer also to [Ste21], where coarse-graining and reconstruction for concentrations as well as the fluxes is developed.
2 Gradient structures 2.1 Gradient systems and the energy-dissipation principle Let us briefly recall what we mean with a gradient system. Following [Mie16], we call a triple (Q, E, R) a gradient system if (1) Q is a closed convex subset of a Banach space X (2) E : Q → R ∞ := R ∪ {∞} is a functional (such as the free energy) (3) R : Q × X → R ∞ is a dissipation potential, which means that for any u ∈ Q the functional R(u, ·) : X → R ∞ is lower semicontinuous (lsc), nonnegative and convex, and it satisfies R(u, 0) = 0.
We define the dual dissipation potential R * : The gradient system is uniquely described by (Q, E, R) or, equivalently by (Q, E, R * ) and, in particular, in this paper we use the second representation. The dynamics of a gradient system can be formulated in different ways as an equation in X, in R or in X * (the dual Banach space of X), respectively: (1) Force balance in X * : 0 ∈ ∂uR(u,u) + DE(u) ∈ X * , (2) Power balance in R: R(u,u) + R * (u, −DE(u)) = − DE(u),u , (3) Rate equation in X:u ∈ ∂ ξ R * (u, −DE(u)) ∈ X.
(Here, ∂ denotes the subdifferential of convex analysis.) Equations (1) and (3) are called gradient flow equation associated with (Q, E, R * ). The equivalent formulations rely on the following fact: Let X be a reflexive Banach space and Ψ : X → R ∞ be a proper, convex and lsc. Then for every ξ ∈ X * and v ∈ X the following five statements, the so-called Legendre-Fenchel-equivalences, are equivalent: Especially the second dynamic formulation, the power balance (2), is interesting for us.
Integrating the power balance (2) in time form 0 to T and using the chain rule for the 2020-12-07 Artur Stephan time-derivative of t → E(u(t)), we get another equivalent formulation of the dynamics of the gradient system, which is called Energy-Dissipation-Balance: (2.1) Equation (EDB) compares the energy of the system at time t = 0 and at time t = T , the difference is described by the total dissipation from t = 0 to t = T . This gives rise to another definition: We define the De Giorgi dissipation functional as

Definition of EDP-convergence
The definition of EDP-convergence for gradient systems relies on the notion of Γ-convergence for functionals (cf. [Dal93]). If Y is a Banach space and I ε : Y → R ∞ we write I ε Γ − → I 0 and I ε Γ ⇀ I 0 for Γ-convergence in the strong and weak topology, respectively. If both holds this is called Mosco-convergence and written as I ε M − → I 0 . For families of gradient systems (X, E ε , R ε ), three different levels of EDP-convergence are introduced and discussed in [DFM18,MMP20], called simple EDP-convergence, EDPconvergence with tilting and contact EDP-convergence with tilting EDP-convergence with tilting is the strongest notion, since it implies the other two notions. Here we will only use the first two notions. For all three notions the choice of weak or strong topology is still to be decided according to the specific problem.
Definition 2.2 (Simple EDP-convergence). A family of gradient structures (Q, E ε , R ε ) is said to EDP-converge to the gradient system (Q, E 0 , R eff ) if the following conditions hold: −→ D 0 . A general feature of EDP-convergence is that, under suitable conditions, solutions u of the gradient flow equationu = ∂ ξ R * eff (u, −DE 0 (u)) of the effective gradient system (X, E 0 , R eff ) are indeed limits of solutions u ε of the gradient flow equationu = A strengthening of simple EDP-convergence is the so-called EDP-convergence with tilting. This notion involves the tilted energy functionals E η ε : Q ∋ u → E ε (u) − η, u , where the tilt η (also called external loading) varies through the whole dual space X * . Definition 2.3 (EDP-convergence with tilting (cf. [MMP20, Def. 2. 14]). A family of gradient structures (Q, E ε , R ε ) is said to EDP-converge with tilting to the gradient system for all η ∈ X * (and similarly for weak Γ-convergence), since the linear tilt u → − η, u is weakly continuous. The main and nontrivial assumption is that additionally , Q) to D η 0 for all η ∈ X * and that this limit D η 0 is given in The main point is that R eff remains independent of η ∈ X * . We refer to [MMP20] for a discussion of this and the other two notions of EDP-convergence.

Gradient system of reaction-diffusion systems
In this section, we present the gradient system (Q, E, R * ε ), which induces the reactiondiffusion system (1.2). In Section 3.2 we derive the gradient flow equation of the gradient system including general tilts of the energy. In Section 3.3 we compute the primal dissipation potential R ε , which is only implicitly given via a infimal-convolution, and the total dissipation functional D η ε , which will be the main object of interest in Section 4. In Section 3, the computations are basically formal; the precise functional analytic setting is presented in Section 4 which also includes the Γ-convergence and EDP-convergence result.
3.1 Gradient structure for the linear reaction system Although a gradient system induces a unique gradient flow equation, a general evolution equation can often be described by many different gradient systems. The choice of the gradient structure is a question of modeling since it adds thermodynamic information to the system, which is not inherent in the evolution equation itself. Here, we follow the pioneering work of Otto and coauthors [JKO98,Ott01] who showed that certain diffusion type equations can be understood as a gradient flow equation of the free energy in the space of probability measures equipped with the Wasserstein distance. Later Mielke proposed a gradient structure for a reaction diffusion system satisfying detailed balance [Mie11]. For a system with two species with a reversible reaction detailed balance is always satisfied. For the reaction part, we use the gradient structure which has been derived via a large-deviation principle from a microscopic Markov process in [MPR14]. We refer also to [Ren18], where our choice of gradient structure has been formally derived.
The gradient system (Q, E, R * ε ) is defined as follows: The state space is the space of probability measure on Q × {1, 2} where we assume that Ω ⊂ R d is a compact domain with normalized mass |Ω| = 1. The driving energy functional E : Q → R ∞ := R ∪ {∞} is the free-energy of the reactiondiffusion system. It is finite for measures µ = (µ 1 , µ 2 ) with Lebesgue density c = (c 1 , c 2 ) only and has the form where the Boltzmann function is defined as E B (r) = r log r − r + 1 and the positive stationary measure is given by w = 1 α+β (β, α) T . Note that the stationary measure w as well as the energy E is ε-independent. The derivative of the energy E is only defined in its domain, i.e. for measures with Lebesgue density c, and has the form As the equation splits into a diffusion and reaction part, so does the dual dissipation functional. We define where we use the cosh-function C * (x) = 4 (cosh(x/2) − 1) and for measures µ with Lebesgue density c we have d √ µ 1 µ 2 := √ c 1 c 2 dx.
The diffusion part R * diff induces the Wasserstein distance on Q. The ε-dependent reaction part R * react,ε forces the evolution close to a linear submanifold given by Note, that since R * react,ε is not 2-homogeneous, it does not define a metric on Q. We refer to [PR * 20] which treats similar and general dissipation potentials and understands them as generalized transport costs on discrete spaces. Note that R * ε does not depend on the stationary measure w explicitly, as highlighted in [MiS20].

The tilted gradient flow equation
In this section, we derive the gradient flow equation of the gradient system (Q, E, R * ε ). To exploit the full information of the dissipation potential, we consider general tilted energies. First, we present how a change of energy by a linear tilt corresponds to a change of stationary measure, and secondly, we compute the induced gradient flow equation.
Let us first consider two free energies (3.1) with different stationary measures w, w, which may be space dependent but are assumed to be positive. Assuming a density µ = c dx and using i Ω w i dx = i Ω c i dx = 1 (where we used |Ω| = 1), we have In particular, we conclude that E(µ) Hence, changing the underlying stationary measure corresponds to a linear tilt of the energy by a two component potential On the other hand, a tilted energy has a different stationary measure as its minimum. To compute the new stationary measure, we introduce tilted energies where V ∈ C 1 (Ω, R 2 ) is a two component smooth potential. Moreover, we introduce η i := e −V i and clearly, we have η i > 0 on Ω ⊂ R d . We compute the stationary state w V by minimizing E V on the space Q = Prob(Ω × {1, 2}). We obtain the space dependent stationary measure Next, we compute the tilted gradient flow equationċ = ∂ ξ R * ε (µ, −DE V (µ)), which is induced by the gradient system (Q, which is a system of two uncoupled drift-diffusion equations or Fokker-Planck equations for the concentrations c i where the fluxes are given by a diffusion part −δ i ∇c i and a drift part −δ i c i ∇V i . For the reaction part of the dual dissipation potential, we insert ). On readily verifies the identity (C * ) ′ (log p − log q) = p−q √ pq for the cosh-function and conclude Hence, we get which is linear in c = (c 1 , c 2 ). In vector notation, we get a tilted Markov generator of the form which has the space dependent stationary measure Summarizing, the tilted evolution equation has the form d dt 3) which is a linear reaction-drift-diffusion system with space dependent reaction rates. In the special case without external forcing V = const, we get the linear reaction diffusion system (1.2). Note that the reaction part still inherits symmetry since the product of the off-diagonal elements is constant in space. In particular, not all general linear reactiondrift-diffusion system with space dependent reaction rates for two species can be expressed in the form (3.3) and are induced by the gradient system (Q,

The dissipation functional
In this section, we compute the dissipation functional D ε , which consists of two parts: the velocity part given by the primal dissipation potential R ε and the slope-part (sometimes also called Fisher information) R * ε (µ, −DE(µ)). Here, all computations are formal and we always assume that the measure µ has a Lebesgue density c. The precise functional analytic setting is presented in the Section 4.
The primal dissipation potential R ε , given by the Legendre transform of the dual dissipation potential R * ε = R * diff + R * ε,react , can be computed via inf-convolution of R diff and R react,ε . First, we compute both primal dissipation potentials separately. To do this, we introduce the following notation: For a convex, lsc. function F : X → [0, ∞] on a reflexive and separable Banach space X with Legendre dual F * , we define the function Introducing the quadratic function Q(x) = 1 2 |x| 2 on R d , the primal dissipation potential of the diffusion part R * diff is given by where J j is, by definition, the unique solution of the elliptic equation δ j c j . The primal dissipation potential of the reaction part is where C = (C * ) * is the Legendre transform of the cosh-function C * (x) = 4 (cosh(x/2) − 1). In the following, we use the inequality which, in particular, implies that the Orlicz class for A ⊂ R d given by A uv dx . Importantly, functions Q, C as well as the functionals R diff , R react,ε are convex on their domain of definition.
The primal dissipation potential R ε is the inf-convolution of R diff and R react,ε , and is given by In time-integrated form we get for v =μ that where we introduce the notation of a (linear) generalized continuity equation Without the reaction part, T 0 R ε dt is the dynamic formulationà la Benamou-Brenier of the Wasserstein distance in Q [BeB00], which can be equivalently written in the form expressed in terms of transport velocities v j = J j /c j . The Wasserstein distance can be interpreted as a cost in transporting mass from one measure µ 0 to µ 1 . In our situation T 0 R ε dt is jointly convex in c, J and b and corresponds to modified cost function which also takes the reaction fluxes into account. The optimal diffusion fluxes J j and reaction fluxes b j have to satisfy the generalized continuity equation. Note that T 0 R ε dt does not induce a metric on Q since the reaction part is not quadratic.
Next, we compute the tilted slope part R * ε (µ, −DE V (µ)). To do this, we introduce the relative densities ρ V of µ w.r.t. the stationary measure EDP-convergence for linear RDS 2020-12-07 Artur Stephan For the reaction part, we use the identity C * (log p − log q) = 2 ( Summarizing, the total dissipation functional is

EDP-convergence result
In this section we state the EDP-convergence result for the gradient systems (Q, E, R * ε ) to (Q, E, R * eff ) and discuss the properties of the effective gradient system (Q, E, R * eff ). Since the energy E is ε-independent the major challenge is to prove Γ-convergence of the dissipation functional D V ε , which is a functional defined on the space of trajectories in the state space Q. To be mathematical precise, we first fix the functional analytic setting.
The state space Q = Prob(Ω×{1, 2}) is equipped with the p-Wasserstein distance d Wp , where in our situation either p = 1 or p = 2. Recall that for any compact euclidean subspace E ⊂ R k the p-Wasserstein distance is defined on the space of probability measures Prob(E) by where Γ(µ 1 , µ 2 ) is the set of all transport plans with marginals µ 1 and µ 2 (see e.g. [AGS05]). The p-Wasserstein distance d Wp metrizises the weak*-topology of measures, i.e. convergence tested against continuous functions on E. In the following we will con- To define the topology in the space of trajectories on Q, we start very coarse, where we understand the trajectories on Q as measures in space and time. We denote the space of trajectories by L ∞ w ([0, T ], Q) equipped with the weak*-measureability. The weak*convergence is defined as usual by A finer topology, which enables to prove compactness and evaluate the effective dissipation functional is then given by the a priori bounds In fact, as presented in Section 5.1, these bounds provide that the measures µ ε have Lebesgue densities c ε which converge strongly in L 1 ([0, T ]×Ω, R 2 ≥0 ). Moreover, the limiting coarse-grained measureμ 0 = µ 0 1 + µ 0 2 has an representative which is absolutely continuous is not a trajectory in the space of probability measure, but in the space of non-negative Radon measures. Proposition 5.11 shows that µ 0 i is absolutely continuous in time with values in (M + (Ω), d W 1 ) exploiting the dual formulation of the 1-Wasserstein distance (see e.g. [Edw11]). This compactness result is comparable to the result of Bothe and Hilhorst [BoH02], where also strong convergence of solutions c = (c 1 , c 2 ) is proved. In particular, similar to the space independent situation in [MiS20, Ste19, MPS20] one cannot guarantee that µ ε (t) → µ 0 (t) in Q for all times t ∈ [0, T ] as jumps in time cannot be excluded. Instead the limit µ 0 = c 0 dx has an absolutely continuous representative.

Main theorem
Let us state our main EDP-convergence result. For doing this, we define for V ∈ C 1 (Ω, R 2 ) the total dissipation functional on L ∞ w ([0, T ], Q) as otherwise.
If µ = c dx a.e. in [0, T ], then the dissipation functional is given by The theorem states that the limit dissipation functional is again of R ⊕ R * -form with an effective dissipation potential R * eff . The effective dissipation potential R * eff consists again of two terms describing the diffusion and a coupling, which forces the chemical potential −DE V to equilibration. This equilibration provides the microscopic equilibria of the densities ρ V defining the slow manifold of the evolution.
In the Section 5, we will present the detailed proof of this Γ-convergence result. In this section, we discuss the effective gradient system and its induced gradient flow equation. As we will see the associated gradient flow equation can be understood as an evolution equation on Q and also on a smaller state spaceQ := Prob(Ω) of coarse-grained variables.
As an immediate consequence, Theorem 4.2 implies that (Q, E, R * ε ) EDP-converges with tilting to (Q, E, R * eff ). Theorem 4.3. Let R * eff be defined by (4.2). Then the gradient system (Q, E, R * ε ) EDPconverges with tilting to (Q, E, R * eff ). Proof. The energy E is ε-independent and lsc. on Q. Hence, it Γ-converges to itself. Theorem 4.2 implies that D V ε Γ-converges to D V 0 and D V 0 is of R ⊕ R * structure, where the effective dissipation potential R eff is independent of the tilts η = V . Hence, EDPconvergence with tilting is established.

Effective gradient flow equation
In this section, we discuss the effective gradient flow equation that is associated by the limit gradient structure (Q, E, R * eff ). Similar to the space-independent situation in [MiS20, MPS20] the limit gradient structure can also be equivalently understood as a gradient structure on a smaller coarse-grained space of slow variablesQ. In particular, we obtain an effective gradient flow equation on the original state space Q with a Lagrange multiplier ensuring the projection on the slow manifold, and, moreover, an effective gradient flow equation in coarse-grained variables. First, we discuss the effective gradient flow equation with Lagrange multipliers, and secondly, the coarse-grained gradient structure and its induced gradient flow equation. Throughout the section the potential V ∈ C 1 (Ω, R 2 ) is fixed.

Gradient flow equation with Lagrange multipliers
For being brief, the calculations in this section are rather formal. The effective dissipation potential R * eff = R * diff + χ {ξ 1 =ξ 2 } consists of two parts: the first describes the dissipation of the evolution and the second provides the linear constraint of being on the slow manifold and also the corresponding Lagrange multiplier. The evolution equation is given bẏ Following [EkT76], the subdifferential of a sum is given by the sum of the subdifferential, if one term is continuous, which holds for the first term. For the second term, the subdifferential of the characteristic function is only definite in its domain, i.e. if which implies that µ = cdx and that their densities satisfy the relation c 1 defining the linear slow manifold. Moreover, on its domain we have for the subdifferential that ∂χ {ξ 1 =ξ 2 } = M(Ω) 1 −1 . Hence, we conclude thaṫ which implies the gradient flow equation on the slow manifold with a Lagrange multiplier λ = (λ 1 , λ 2 ) for the densities of the form

Coarse-grained gradient structure and its gradient flow equation
Now, we discuss the effective gradient structure (Q, E, R * eff ) in the slow coarse-grained variables. To do this, we introduce the coarse grained probability measureμ = µ 1 + µ 2 on Ω and the corresponding concentrationsĉ := c 1 + c 2 . Moreover, we define the equilibrated densitiesρ V = ρ V 1 = ρ V 2 and the coarse-grained stationary measureŵ V = w V 1 + w V 2 , for EDP-convergence for linear RDS 2020-12-07 Artur Stephan With this notation, we may define the coarse-grained gradient structure (Q,Ê,R * ). On the state spaceQ = Prob(Ω), we definê Introducing the coarse-grained potentialV = − log(w , for which the exponential is given by the weighted arithmetic mean of the exponentials e −V 1 and e −V 2 , i.e. e −V = w 1 e −V 1 + w 2 e −V 2 (we used that w 1 + w 2 = 1). Easy calculations show that the energy has for the explicit form The coarse-grained dissipation functional is defined bŷ which incorporates the tilt via the coarse-grained variables. Note, that the coarse-grained dissipation potentialR * depends explicitly on the tilt V via the diffusion coefficient δ V . This is not a contradiction to tilt-EDP convergence (Theorem 4.3), because in original variables the effective dissipation potential (4.3) is indeed independent of the tilts. The tilts dependence ofR * originates from the energy and tilt dependent slow manifold.
To relate the dissipation functional D V 0 with the coarse grained dissipation functional D, we first show that also an equilibration of the fluxes occurs. To do this the following convexity property is important.
Lemma 4.4. Let X a separable and reflexive Banach space and let F : X → R ∞ be convex and lsc. Then for the function F : If F is strictly convex then equality holds if and only if (a i , x i ) = (0, 0) whenever a i = 0 or x i /a i = x j /a j whenever a i , a j > 0. Moreover, if F(0) = 0, we have the following monotonicity property F(a 1 , x) ≤ F(a 2 , x), if a 1 ≥ a 2 .
Proof. Let pairs (a i , x i ) for i = 1, . . . , I be given. If a i = 0, then either x i = 0 and the claim has to be shown for I − 1 -number of pairs, or x i = 0 and the right-hand side is ∞ meaning that the claim is trivial. So let us assume that a i > 0 for all i = 1 . . . , I. Then F(a i , x i ) = a i F(x i /a i ) and the claim is equivalent to which holds since F is convex. If F is strictly convex then we immediately observe that whenever a i , a j > 0 we have x i a i = x j a j , and whenever a i = 0 that also x i = 0. To see the monotonicity property, we observe that F(a, 0) = 0 for all a ≥ 0. Hence, we have F(a 1 + a 2 , x) ≤ F(a 1 , x) + F(a 2 , 0) = F(a 1 , x), which proves the claim.
Recalling formula (4.3) of the effective dissipation functional D V 0 and using the above lemma, we observe that the velocity part of the dissipation functional D V 0 can now be estimated. In particular, we will see that the limit dissipation functional D V 0 can be equivalently expressed in coarse-grained variables (μ,Ĵ) by using that an equilibration of concentrations also provides an equilibration of the corresponding fluxes. In the reconstruction strategy in Section 5.3 this equilibration is explicitly used (see 4.10 and (5.3)).
3. The gradient flow equation of the gradient system (Q,Ê,R * ) is given bẏ with the potentialV = − logŵ V and stationary measureŵ V .
Equation (4.6) shows that the coarse-grained gradient flow equation induced by (Q,Ê,R * ) is a drift-diffusion equation of the coarse-grained concentrationĉ with mixed diffusion constantδ V . In particular, in the tilt free case we haveδ V =const = βδ 1 +αδ 2 α+β , and we recover the result of [BoH02].
Proof. To prove Part 1, we first observe that the bounded energy and dissipation for the trajectory µ implies that we have c 1 a.e. in [0, T ] × Ω. Usingĉ = c 1 + c 2 for the densities, we getĉ = (4.8) Lemma 4.4 provides that also an equilibration of the fluxes occurs. Indeed, defining the coarse-grained fluxĴ = J 1 + J 2 , we conclude Equality holds if and only if (J 1 , c 1 ) = 0 or (J 2 , c 2 ) = 0 or which provides an explicit formula for the coarse-grained diffusion flux.
For the dissipation functional that means To prove equality, we first observe thatĴ , J 1 , J 2 satisfy the same boundary conditions. Moreover, the explicitly derived reaction flux b 1 , b 2 from (5.3) shows that the reconstructed fluxes (J 1 , J 2 ) from coarse-grained fluxĴ is admissible. Hence, we obtain equality which proves the first part. For Part 3, we compute the evolution equation that is induced by the gradient system is (Q,Ê,R * ). We have ∂ξR * (μ,ξ) = −div δ Vĉ ∇ξ , Note that the coarse-grained gradient flow equation (4.6) is equivalent to the gradient flow equation with Lagrange multipliers (4.4). Indeed, adding both equations in (4.4) together and using that the original concentrations can be expressed by the coarse-grained concentrations via the coarse-grained gradient flow equation (4.6) with the drift termδ Vĉ ∇V can be readily derived. Conversely, using (4.12), we see that c = (c 1 , c 2 ) are on the slow manifold and satisfy (4.4). The corresponding Lagrange multipliers λ = (λ 1 , λ 2 ) can be explicitly calculated. Introducing the difference of the diffusion constants δ = δ 1 − δ 2 and the potentials V = V 1 − V 2 , we have We observe that the Lagrange multiplier λ i has the same regularity as the right-hand side of the evolution of c i . Moreover, both evolution equations are completely uncoupled but contain a linear annihilation/creation term, which depends on the potential V = (V 1 , V 2 ) and the diffusion coefficient δ = (δ 1 , δ 2 ). A lengthy calculation shows that indeed we have λ 1 + λ 2 = 0.

Proof of Γ-convergence
In this section we prove the Γ-convergence result of Theorem 4.2. As usual, we prove Γconvergence in three steps: First deriving compactness, secondly establishing the liminfestimate by exploiting the compactness, thirdly constructing the recovery sequence for the limsup-estimate.
In the following the next lemma will be useful. Proof. Let W and ρ be given. We define three measurable subsets of Ω: Since F is superlinear, there is a constant k F > 0 such that on Ω 2 it holds W/ρ ≤ k F . Hence we can estimate Moreover, we need the following classical lemma. It guarantees the necessary regularity for the limits, and moreover, it provides the desired liminf-estimate. otherwise.

Compactness
In this section, we derive the required compactness for proving the liminf-estimate in Section 5.2. Recall that for given potential V ∈ C 1 (Ω, R 2 ) the dissipation functional D V ε is defined on the space of trajectories equipped with the weak topology, i.e. µ ε → µ 0 ∈ L ∞ w ([0, T ], Q) if and only if it holds In the following we want to derive compactness for a sequence (µ ε ) ε>0 of trajectories, satisfying the a priori bounds sup ε>0 ess sup where the total dissipation functional is Using the bound of the dissipation functional D V ε (µ ε ) ≤ C, we conclude that there are diffusive fluxes J ε = (J ε 1 , J ε 2 ) and reaction fluxes b Moreover, we get bounds: Remark 5.3. Following [Man07] a distributional solution (µ, J, B) of the generalized continuity equationμ = −divJ + B satisfying T 0 Ω |B| + |J|dx < ∞ can be assumed to be absolutely continuous. The bounds can be obtained easily using Lemma 5.1 for fixed ε > 0.
Although the functional is convex in the concentration c and in the fluxes J and b, weak convergence would be sufficient to prove a liminf-estimate using a Joffe-type argument. But, comparing the situation with the evolution equation, we aim in proving even strong convergence for the densities c ε → c 0 in L 1 ([0, T ]×Ω, R 2 ≥0 ). This is done in two steps: First, compactness of coarse-grained variables, and secondly, convergence towards the slow manifold is shown, which together implies strong compactness. This strategy has successfully been applied already in the space-independent case in [MiS20,MPS20]. Moreover, we show that the limit trajectory µ 0 = c 0 dx has a representative which is in AC([0, T ], Q). Note that it is not possible to prove pointwise convergence µ ε (t) * ⇀ µ 0 (t) for all t ∈ [0, T ]. Instead, pointwise convergence is only shown for the coarse-grained variablesμ ε := µ ε 1 + µ ε 2 . First, we derive weak compactness in space-time, which immediately follows from the uniform bound in ε and time on the energy.  Proof. The bound on the energy implies that a.e. t ∈ [0, T ] the measure µ ε (t, ·) has a Lebesgue density c ε (t, ·). Moreover, the functional µ → T 0 E(µ)dt is superlinear and convex. Hence, it follows by the Theorem of de Valleé -Poussin that µ ε are uniformly integrable and hence, µ ε i converges weakly in L 1 ([0, T ] × Ω) to µ 0 i . In the following, we are going to derive compactness for the concentrations c ε i and the diffusive fluxes J ε i . It is not possible to get compactness for the fast reaction flux b ε 2 by bounding the dissipation functional. In particular, pointwise convergence for the measures µ ε (t) cannot be achieved.
Remark 5.5. To see that compactness for the fast reaction flux b ε 2 is not possible obtain, Then, a bound means on the dissipation functional implies a bound Setting b ε = − log ε, we easily see that |b ε | log(ε|b ε | + 1) → 0 as ε → 0, however, b ε → ∞. Hence, it is not possible to obtain compactness for the fast reaction flux b ε i . Later in Lemma 5.14 the "converse" statement is proved: Next, we are going to derive time-regularity for the sequence (µ ε ) ε>0 in proving compactness for the coarse-grained trajectoriesμ ε = µ ε 1 + µ ε 2 . In particular, we are able to prove pointwise convergence in time.
Next, the compactness result from Lemma 5.2 is used in order to prove compactness for the fluxes and spatial regularity.
Proof. By the bound on the dissipation functional, we get (after extracting a suitable . Hence applying the Lemma 5.2, we conclude that J 0 j ≪ µ 0 j . Similarly, we conclude compactness for the gradients ∇ρ V,ε . The only thing that remains is to identify the limit. But this is clear by definition of the weak derivatives, i.e. integrating against smooth test functions, because this is captured in the weak star convergence. Lemma 5.1 implies that ρ ε j is uniformly bounded in L 1 ([0, T ], W 1,1 (Ω)). The spatial regularity and the temporal regularity provides a compactness result by a BV-generalization of the Aubin-Lions-Simon Lemma.
In our situation we immediately conclude thatĉ ε converges strongly.
It is also clear that we get convergence towards the fast manifold, which results from the Fisher information of the fast reaction. Proof. The bound on the dissipation functional provides Cε. Hence, we conclude ρ V,ε 1 − ρ V,ε 2 L 2 ([0,T ]×Ω) → 0 as ε → 0. In particular, we conclude that ρ V,0 1 = ρ V,0 2 . The strong convergence towards the slow manifold provides strong convergence for the whole sequence. Indeed, using Cauchy-Schwartz inequality and The last term can be estimated by the AM-GM inequality dxdt, and the right-hand side is bounded since µ(t) ∈ Q for t ∈ [0, T ]. Hence, we conclude that and both terms converge strongly to zero as ε → 0 by convergence ofĉ ε →ĉ 0 and ρ V,ε 1 − ρ V,ε 2 → 0. Finally, we show that the limit µ 0 = c 0 dx has an absolutely continuous representative in the space of probability measures. To do this, we exploit the characterization of absolutely continuous curves as solutions of the continuity equation following [AGS05].
Proposition 5.11. Let (µ ε ) ε>0 , µ ε ∈ L ∞ ([0, T ], Q) satisfying the a priori bounds (5.1) and let c 0 be the limit of the densities c ε . Then the coarse-grained slow variableμ = µ 0 ) has a representative (in time), which is absolutely continuous in the space of probability measures equipped with the 2-Wasserstein metric. Moreover, each component µ 0 i has an absolutely continuous representative (in time), which is absolutely continuous in the space of non-negative Radon measures equipped with the 1-Wasserstein metric.
Proof. The coarse-grained measuresμ ε satisfy continuity equationμ ε + div(Ĵ ε ) = 0 in the sense of distributions whereĴ ε = J ε 1 + J ε 2 is the coarse-grained diffusion flux. Since the linear continuity equation is stable under weak convergence, we conclude that also the limits satisfy the same continuity equationμ 0 + div(Ĵ 0 ) = 0, whereĴ 0 is the weak*-limit ofĴ ε (see Lemma 5.7). Using (4.9), the bound on the dissipation functional implies a bound Ω |v| 2 dμdt and the bound on the dissipation functional implies the bound on the Borel velocity field v L 2 (μ) < ∞. Hence, by Theorem 8.3.1 from [AGS05] it follows that t →μ(t) ∈ (Prob(Ω), d W 2 ) has a continuous representative which is absolutely continuous.
To prove time-regularity for µ 0 i for i = 1, 2, we first observe that is a nonnegative Radon measure. To show that it has an absolutely continuous representative, we proceed as in Lemma 5.6 and exploit the dual formulation of the 1-Wasserstein distance on the space of non-negative Radon measures, i.e. integrating against Lipschitz functions (see e.g. [Edw11]). Let φ ∈ C 1 (Ω) with φ W 1,∞ (Ω) ≤ 1. Using the continuity equationṡ where C = C(w, V ). The bound on the dissipation functional provides again that the right-hand side is bounded for each interval [t 1 , t 2 ] ⊂ [0, T ]. Summing up, we conclude that µ 0 i has an absolutely continuous representative, which proves the claim.

Liminf-estimate
In this section, we state and prove the liminf-estimate of the Γ-convergence result Theorem 4.2. Once the compactness is established the proof of the liminf-estimate is comparatively easy.

Construction of the recovery sequence
In this section, we construct the recovery sequence for the functional D V 0 to finish the Γ-convergence result in Theorem 4.2. To be precise, we will show the following: ) such that the a priori bounds D V 0 (µ 0 ) < ∞ and ess sup t∈[0,T ] E(µ 0 (t)) < ∞ hold. Then there is a sequence (µ ε ) ε>0 , µ ε ∈ AC([0, T ], Q), sup ε>0 ess sup t∈[0,T ] E(µ ε (t)) < ∞, such that the densities converge c ε → c 0 strongly in Using Proposition 4.5, we see that the limit functionals do not contain more information than the functionals in coarse-grained variables and it holds D V 0 (µ 0 ) =D(μ 0 ). Hence, we may reconstruct the dissipation functional with the corresponding diffusion and reaction flux (J, b) from the coarse-grained variables (ĉ,Ĵ).
The dissipation functional D V 0 is defined on the space of general fluctuations around the solution of the evolution equation (1.2). These fluctuations are neither strictly positive nor smooth. The proof of the limsup-estimate is done in several steps, which are elaborated in the next lemmas. The bound D V 0 (µ 0 ) < ∞ can be assumed without loss of generality because the other case is already treated in the liminf-estimate.
Hence, defining the recovery sequence µ ε := µ ǫ,γ , we have where the first term tends to zero by the first reconstruction step, the second term tends to zero by the third reconstruction step and the third term tends to zero by the second reconstruction step, which in total proves the desired convergence.
Before performing the three recovery steps in Section 5.3.2, we first illustrate the general idea of constructing the recovery sequence by forgetting about positivity and regularity issues for the first moment.

Construction of recovery sequence for smooth and positive measures
To show the general idea, let us firstly assume that the density ofμ is sufficiently smooth and positive, i.e. we assume that its Lebesgue density satisfiesĉ ≥ 1 C > 0 on Ω × [0, T ] and has a enough regularity that will be specified below. LetĴ be the diffusion flux which EDP-convergence for linear RDS 2020-12-07 Artur Stephan provides the minimum inD(μ 0 ) = D V 0 (µ) and satisfiesċ + div(Ĵ) = 0. We define the reconstructed variables by The reconstructed concentrations c and diffusion fluxes J = (J 1 , J 2 ) are proportional to the coarse grained concentrationĉ and diffusion fluxĴ , respectively. On the coarsegrained level, which considers only one species there is no reaction flux anymore. (This changes when considering large reaction-diffusion system as explained in Section 6). The reactive flux b = (b 1 , b 2 ) is given as a function of the coarse-grained diffusion fluxĴ, which means that in the limit the diffusion determines the hidden reaction.
Concerning regularity issues, we immediately observe the following. Since w V is smooth and positive, c 1 , c 2 have the same regularity asĉ and also J 1 , J 2 have the same regularity asĴ . Only the reaction fluxes b i are a priori not well-defined (e.g. in L 1 ) if divĴ andĴ are not regular enough. This means, that the reaction flux between the fast-connected species is not well-defined for generalĴ ∈ M(Ω × [0, T ], R 3 ). Note, that regularity assumptions for divĴ are not needed if δ 1 = δ 2 , i.e. if both species diffuse with the same diffusion constant. In particular, in this situation no regularization argument as in Lemma 5.20 is necessary. Moreover, no additional regularity forĴ is needed if which means that the potentials V 1 , V 2 differ in a constant on Ω. In particular, this implies that for the coarse-grained potentialV we have ∇V = ∇V 1 = ∇V 2 . As we will see, enough regularity forĴ is already obtained from bounds on the dissipation functional. Of course, regularity properties for divĴ andĴ are independent of each other. So let us assume for the moment that b i is well-defined. Then we conclude (c, J, b) ∈ (gCE), because we havė where we used that (ĉ,Ĵ) solvesċ + divĴ = 0. Similarly, we see thatċ 2 + divJ 2 = b 2 and, by definition, we have b 1 + b 2 = 0. Moreover, boundary properties ofĴ remain for J = (J 1 , J 2 ). Since c 1 , we conclude that D V ε (µ) ≤ D 0 (µ) + T 0 Ω C √ c 1 c 2 ε , b 2 dxdt. That means, that for proving that the constant sequence µ ε = µ is a recovery sequence, it T 0 Ω C √ c 1 c 2 ε , b 2 dxdt → 0 as ε → 0. This is, in fact, shown in the next lemma under the assumption that divĴ,Ĵ ∈ L C ([0, T ] × Ω). The proof basically uses the monotonicity property of the Legendre dual function C(a 1 , b) ≤ C(a 2 , b) as a 1 ≥ a 2 (see Lemma 4.4), its superlinear growth and the dominated convergence theorem.
In fact, the above proof is quite robust and already suggests that the same convergence holds even ifĉ ε → 0 not to fast somewhere in Ω × [0, T ] and b 2 ε ≈ ε −α for some α > 0. This is proved in Proposition 5.21.
In the following, we need to overcome the positivity assumptionĉ ≥ 1 C and the regularity assumption forĴ and divĴ. The first is done by shifting the densityĉ δ := 1 Z δ (ĉ + δ), δ > 0; the necessary regularity ofĴ is provided immediately by the bound on the dissipation functional; the regularity of divĴ is achieved by smoothing using that (ĉ,Ĵ) is a solution of the coarse-grained continuity equationċ + divĴ = 0.

Auxiliary results for construction of recovery sequence for general measures
First, we show how to overcome the positivity assumption. This is done by a controlled positive shift.
With these preparations, we are able to show the remaining step in the proof of Theorem 5.13. The next proposition shows in analogy to Lemma 5.14 that the contribution by the reaction flux b ǫ 2 to the dissipation functional D V ε converges to zero as ǫ → 0.