From diffusion to reaction via Gamma-convergence
FFrom diffusion to reaction via Γ-convergence
Mark A. Peletier ∗ Giuseppe Savar´e † Marco Veneroni ‡ November 6, 2018
Abstract
We study the limit of high activation energy of a special Fokker-Planck equation, knownas Kramers-Smoluchowski (K-S) equation. This equation governs the time evolution of theprobability density of a particle performing a Brownian motion under the influence of achemical potential
H/ε . We choose H having two wells corresponding to two chemical states A and B . We prove that after a suitable rescaling the solution to (K-S) converges, in the limitof high activation energy ( ε → A and B , and the reaction A (cid:10) B .The aim of this paper is to give a rigorous proof of Kramer’s formal derivation and toembed chemical reactions and diffusion processes in a common variational framework whichallows to derive the former as a singular limit of the latter, thus establishing a connectionbetween two worlds often regarded as separate.The singular limit is analysed by means of Gamma-convergence in the space of finite Borelmeasures endowed with the weak- ∗ topology. Key words and phrases: unification, scale-bridging, upscaling, high-energy limit, acti-vation energy, Dirichlet forms, Mosco-convergence, variational evolution equations
AMS subject classification:
In a seminal paper in 1940, Hendrik Anthony Kramers described a number of approaches tothe problem of calculating chemical reaction rates [12]. One of the limit cases in this paper isequivalent to the motion of a Brownian particle in a (chemical) potential landscape. In thisdescription a reaction event is the escape of the particle from one energy well into another.This description is interesting for a number of reasons. It provides a connection betweentwo processes, diffusion and reaction, which are often—especially at the macroscopic level—viewed as completely separate. It also provides a link between a macroscopic effect—chemicalreaction—and a more microscopic, underlying motion, and in doing so, it highlights the factthat diffusion and reaction ultimately spring from the same underlying motion. It finally alsoallows for explicit calculation of reaction rates in terms of properties of the energy landscape.In this paper we contribute to this discussion by studying the limit process of high acti-vation energy in the unimolecular reaction A (cid:10) B . As a first contribution, this provides arigorous proof of the result that Kramers had derived formally. At the same time we extendhis result to a Brownian motion in the product space spanned by both the chemical variableof Kramers and the variables corresponding to position in space, resulting in a limit systemthat models not only chemical reaction but also spatial diffusion—a simple reaction-diffusionsystem. ∗ Department of Mathematics and Institute for Complex Molecular Systems, Technische Universiteit Eindhoven,The Netherlands, [email protected] † Dipartimento di Matematica F.Casorati, Universit`a degli studi di Pavia, Italy, [email protected] ‡ Fakult¨at f¨ur Mathematik, Technische Universit¨at Dortmund, Germany, [email protected] a r X i v : . [ m a t h . A P ] D ec ith this paper we have two aims. The first is to clarify the mathematical—rigorous—aspects of the formal results of [12], and extend them to include spatial diffusion, and in thisway to contribute to the upscaling of microscopic systems. The second is to make a first stepin the construction of a variational framework that can describe the combination of generaldiffusive and chemically reactive processes. From this point of view it would be interesting,for example, to place the limit system in the context of Wasserstein gradient flows (see alsoSection 1.10). Initiated by the work of Otto [11, 15] and extended into many directions since,this framework provides an appealing variational structure for very general diffusion processes,but chemical reactions have so far resisted representation in the Wasserstein framework.In this paper we only treat the simple equation A (cid:10) B , but we plan to extend the approachto other systems in the future (see also [14]). We consider the unimolecular reaction A (cid:10) B . In chemical terms the A and B particles aretwo forms of the same molecule, such that the molecule can change from one form into theother. A typical example is a molecule with spatial asymmetry, which might exist in twodistinct, mirror-image spatial configurations; another example is that of enzymes, for whichthe various spatial configurations also have different biological functions. Remark.
Classical, continuum-level modelling of the system of A and B particles that diffuseand react (see e.g. [9, 3]) leads to the set of differential equations, where we write A and B for the concentrations of A and B particles: ∂ t A − D ∆ A = k ( B − A ) (1a) ∂ t B − D ∆ B = k ( A − B ) . (1b)(See Section 1.10 for the equal reaction rates). This system will arise as the upscaling limit(see Theorem 1) of the system that we now develop in detail.We next assume that the observed forms A and B correspond to the wells of an appro-priate energy function. Since it is common in the chemical literature to denote by ‘enthalpydifference’ the release or uptake of heat as a particle A is converted into a particle B , we shalladopt the same language and consider the A and B states to correspond to the wells of anenthalpy function H .While the domain of definition of H should be high-dimensional, corresponding to themany degrees of freedom of the atoms of the molecule, we will here make the standardreduction to a one-dimensional dependence. The variable ξ is assumed to parametrize animaginary ‘optimal path’ connecting the states A to B , such that ξ = − A and ξ = 1 to B . Such a path should pass through the ‘mountain pass’, the point whichseparates the basins of attraction of A and B , and we arbitrarily choose that mountain passto be at ξ = 0, with H (0) = 1. We also restrict ξ to the interval [ − , H is showed in Figure 1. − ξH Figure 1: A typical function H Specifically we make the following assumptions about H : H ∈ C ∞ ([ − , H is even in ξ , maximal at ξ = 0 with value 1, and minimal at ξ = ± H ( ξ ) > − < ξ < H (cid:48) ( ± ∓ ) = 0. The assumption of equal depth for the two wells corresponds to anassumption about the rate constants of the two reactions; we comment on this in Section 1.10. .3 Diffusion in the chemical landscape This newly introduced ‘chemical variable’ ξ should be interpreted as an internal degree offreedom of the particle, associated with internal changes in configuration. In the case of twoalternative states of a molecule, ξ parametrizes all the intermediate states along a connectingpath.In this view the total state of a particle consists of this chemical state ξ together withthe spatial position of the particle, represented by a d -dimensional spatial variable x in aLipschitz, bounded, and open domain Ω ⊂ R d , so that the full state space for the particle isthe closure D of D := Ω × ( − ,
1) with variables ( x, ξ ) . Taking a probabilistic point of view, and following Kramers, the motion of the particle willbe described in terms of its probability density ρ ∈ P ( D ), in the sense that for Borel sets X ⊂ Ω and Ξ ⊂ [ − ,
1] the number ρ ( X × Ξ) is the probability of finding the particle at aposition x ∈ X and with a ‘chemical state’ ξ ∈ Ξ.The particle is assumed to perform a Brownian motion in D , under the influence of thepotential landscape described by H . This assumption corresponds to the ‘large-friction limit’discussed by Kramers. The time evolution of the probability distribution ρ then is given bythe Kramers-Smoluchowski equation ∂ t ρ − ∆ x ρ − τ ∂ ξ ` ∂ ξ ρ + ρ ∂ ξ H ´ = 0 in D (cid:48) ( D × (0 , ∞ )), (2)with Neumann boundary conditions on the lateral boundary ∂D . The coefficient τ > x and ξ : since x is a rescaled physicaldistance, and ξ is a rescaled ‘chemical’ distance, the units of length in the two variables aredifferent, and the parameter τ can be interpreted as the factor that converts between the twoscales. Below we shall make an explicit choice for τ . In the setup as described above, there is a continuum of states (i.e. ( − , A state to the B state, and a statement of the type ‘the particle is in the A state’ is therefore notwell defined. In order to make a connection with the macroscopic description ‘ A (cid:10) B ’, whichpresupposes a clear distinction between the two states, we take the limit of high activationenergy, as follows.We rescale the enthalpy H with a small parameter ε , to make it H ( ξ ) /ε . (This is called‘high activation energy’ since max ξ H ( ξ ) /ε = 1 /ε is the height of the mountain that a particlehas to climb in order to change states).This rescaling has various effects on the behaviour of solutions ρ of (2). To illustrate oneeffect, let us consider the invariant measure γ ε , the unique stationary solution in P ( D ) of (2): γ ε = λ Ω ⊗ ˜ γ ε , λ Ω := 1 L d (Ω) L d | Ω , ˜ γ ε = Z − ε e − H/ε L | [ − , (3)(where L , L d are the 1- and d -dimensional Lebesgue measures). The constant Z ε is fixed bythe requirement that γ ε ( D ) = ˜ γ ε ([ − , − ξ ˜ γ ε O (1 / √ ε ) Figure 2: The density ˜ γ ε Since H is strictly positive at any − < ξ <
1, the exponential exp( − H ( ξ ) /ε ) vanishes atall ξ except for ξ = ±
1; therefore the measure γ ε concentrates on the lines ξ = − ξ = 1, nd converges weakly- ∗ as ε → γ given by γ = λ Ω ⊗ ˜ γ, ˜ γ := 12 ` δ − + δ ´ . (4)Here weak- ∗ convergence is to be interpreted in the duality with continuous functions in D (thus considering P ( D ) as a weakly- ∗ closed convex subset of the space M ( D ) = ` C ( D ) ´ (cid:48) of signed Borel measures with finite total variation) i.e.lim ε → Z D φ ( x, ξ ) d γ ε = Z D φ ( x, ξ ) d γ ( x, ξ ) = 12 Z Ω ` φ ( x, − φ ( x, ´ d λ Ω ( x ) , for any φ ∈ C ( D ) . We should interpret the behaviour of γ ε as follows. In the limit ε →
0, the deep wells at ξ = ± ε >
0, there is a positive probability that a particle switches from one well to the otherin any given period of time. The rate at which this happens is governed by the local structureof H near ξ = ± ξ = 0, and becomes very small—of order ε − exp( − /ε ), as weshall see below.In the limit ε = 0, the behaviour of particles in the ξ -direction is no longer recognizableas diffusional in nature. In the ξ -direction a particle can only be in one of two states ξ = ± A and B states. Of the diffusional movement in the ξ -direction only a jump process remains, in which a particle at ξ = − ξ = 1, or vice versa. Since the jumping (chemical reaction) rate at finite ε > ε − exp( − /ε ), thelimiting reaction rate will be zero unless we rescale the system appropriately. This requiresus to speed up time by a factor of ε exp(1 /ε ). At the same time, the diffusion rate in the x -direction remains of order 1 as ε →
0, and the rescaling should preserve this. In order toobtain a limit in which both diffusion in x and chemical reaction in ξ enter at rates that areof order 1, we use the freedom of choosing the parameter τ that we introduced above.We therefore choose τ equal to τ ε := ε exp(1 /ε ) , (5)and we then find the differential equation ∂ t ρ ε − ∆ x ρ ε − τ ε ∂ ξ ( ∂ ξ ρ ε + ε ρ ε ∂ ξ H ) = 0 in D (cid:48) ( D × (0 , ∞ )), (6)which clearly highlights the different treatment of x and ξ : the diffusion in x is independentof τ ε while the diffusion and convection in the ξ -variable are accelerated by a factor τ ε . As is already suggested by the behaviour of the invariant measure γ ε , the solution ρ ε willbecome strongly concentrated at the extremities {± } of the ξ -domain ( − , ρ ε as a family ρ ε ( t, · ) of time-dependent measures , insteadof functions. It turns out that the densities u ε ( t, · ) u ε ( t, · ) := d ρ ( t, · )d γ ε of ρ ε ( t, · ) with respect to γ ε also play a crucial role and it is often convenient to have bothrepresentations at our disposal, freely switching between them. In terms of the variable u ε equation (6) becomes ∂ t u ε − ∆ x u ε − τ ε ( ∂ ξξ u ε − ε ∂ ξ H∂ ξ u ε ) = 0 in (0 , + ∞ ) × D, (7)supplemented with the boundary conditions ∂ ξ u ε ( t, x, ±
1) = 0 for all x ∈ Ω , ∇ x u ε ( t, x, ξ ) · n = 0 on ∂ Ω × [ − , , t > . (8) e choose an initial condition u ε (0 , x, ξ ) = u ε ( x, ξ ) , for all ( x, ξ ) ∈ D, with ρ ε = u ε γ ε ∈ P ( D ) . (9)Let us briefly say something about the functional-analytic setting. It is well known (seee.g. [7]) that the operator A ε := − ∆ x − τ ε ∂ ξξ + ( τ ε /ε ) H (cid:48) ∂ ξ with Neumann boundary condi-tions (8) has a self-adjoint realization in the space H ε := L ( D ; γ ε ). Therefore the weak formof equation (7) can be written as b ε ( ∂ t u ( t ) , v ) + a ε ( u ( t ) , v ) = 0 for all v ∈ V ε , (10)where the bilinear forms a ε and b ε are defined by b ε : H ε × H ε → R , b ε ( u, v ) := Z D u v d γ ε , and V ε := W , ( D ; γ ε ) := n u ∈ L ( D,γ ε ) ∩ W , ( D ) : Z D |∇ x,ξ u | d γ ε < + ∞ o ,a ε : V ε × V ε → R , a ε ( u, v ) := Z D A ε u v d γ ε = Z D “ ∇ x u ∇ x v + τ ε ∂ ξ u ε ∂ ξ v ” d γ ε . Since V ε is densely and continuously imbedded in H ε , standard results on variational evolutionequations in an Hilbert triplet (see e.g. [13, 6]) and their regularizing effects show that a uniquesolution exists in C ([0 , ∞ ); H ε ) ∩ C ∞ ((0 , ∞ ); V ε ) for every initial datum u ε ∈ H ε . ρ ε and u ε The following theorem is the first main result of this paper. It states that for every time t ≥ ρ ε ( t ) solutions of (6) weakly- ∗ converge to a limiting measure ρ ( t ) in P ( D ), whose density u ( t ) = d ρ ( t )d γ is the solution of the limit system (1). Note that for afunction u ∈ L ( D, γ ) the traces u ± = u ( · , ± ∈ L (Ω) are well defined (in fact, the map u (cid:55)→ ( u − , u + ) is an isomorphism between L ( D, γ ) and L (Ω , λ Ω ; R )) . We state our result in a general form, which holds even for signed measures in M ( D ). Theorem 1.
Let ρ ε = u ε γ ε ∈ C ([0 , + ∞ ); M ( D )) be the solution of (6–9) with initialdatum ρ ε . If sup ε> Z D | u ε | d γ ε < + ∞ (11) and ρ ε weakly- ∗ converges to ρ = u γ = 12 u − λ Ω ⊗ δ − + 12 u λ Ω ⊗ δ +1 as ε ↓ , (12) then u ∈ L ( D,γ ) , u , ± ∈ L (Ω) , and for every t ≥ the solution ρ ε ( t ) weakly- ∗ converge to ρ ( t ) = u ( t ) γ = 12 u − ( t ) λ Ω ⊗ δ − + 12 u + ( t ) λ Ω ⊗ δ +1 , (13) whose densities u ± belong to C ([0 , + ∞ ); L (Ω)) ∩ C ((0 , + ∞ ); W , (Ω)) and solve the system ∂ t u + − ∆ x u + = k ( u − − u + ) in Ω × (0 , + ∞ ) (14a) ∂ t u − − ∆ x u − = k ( u + − u − ) in Ω × (0 , + ∞ ) (14b) u ± (0) = u , ± in Ω . (14c) The positive constant k in (14a,b) can be characterized as the asymptotic minimal transitioncost k = 1 π p | H (cid:48)(cid:48) (0) | H (cid:48)(cid:48) (1) = lim ε ↓ min n τ ε Z − ` ϕ (cid:48) ( ξ ) ´ d˜ γ ε : ϕ ∈ W , ( − , , ϕ ( ±
1) = ± o . (15) emark (The variational structure of the limit problem) . The “ ε = 0” limit problem (14a-14c) admits the same variational formulation of the “ ε >
0” problem we introduced inSection 1.6. Recall that γ is the measure defined in (4) as the weak limit of γ ε ; we set H := L ( D, γ ), and for every ρ = uγ with u ∈ H we set u ± ( x ) := u ( x, ± ∈ L (Ω , λ Ω ). Wedefine b ( u, v ) := Z D u ( x, ξ ) v ( x, ξ ) d γ ( x, ξ ) = 12 Z Ω “ u + v + + u − v − ” d λ Ω . (16)Similarly, we set V := ˘ u ∈ H : u ± ∈ W , (Ω) ¯ , which is continuously and densely imbeddedin H , and a ( u, v ) := 12 Z Ω “ ∇ x u + ∇ x v + + ∇ x u − ∇ x v − + k ` u + − u − ´ ( v + − v − ) ” d λ Ω . (17)Then the system (14a,b,c) can be formulated as b ( ∂ t u ( t ) , v ) + a ( u ( t ) , v ) = 0 for every t > v ∈ V , (18)which has the same structure as (10). u ε Weak- ∗ convergence in the sense of measures is a natural choice in order to describe thelimit of ρ ε , since the densities u ε and the limit density u = ( u + , u − ) are defined on differentdomains with respect to different reference measures. Nonetheless it is possible to consider astronger convergence which better characterizes the limit, and to prove that it is satisfied bythe solutions of our problem.This stronger notion is modeled on Hilbert spaces (or, more generally, on Banach spaceswith a locally uniformly convex norm), where strong convergence is equivalent to weak con-vergence together with the convergence of the norms: x n → x ⇐⇒ x n (cid:42) x and (cid:107) x n (cid:107) → (cid:107) x (cid:107) . (19)In this spirit, the next result states that under the additional request of “strong” convergenceof the initial data u ε , we have “strong” convergence of the densities u ε ; we refer to [17, 10](see also [2, Sec. 5.4]) for further references in a measure-theoretic setting. Theorem 2.
Let ρ ε , ρ ε be as in Theorem 1. If moreover lim ε ↓ b ε ( u ε , u ε ) = b ( u , u ) , (20) then for every t > we have lim ε ↓ b ε ( u ε ( t ) , u ε ( t )) = b ( u ( t ) , u ( t )) (21) and lim ε ↓ a ε ( u ε ( t ) , u ε ( t )) = a ( u ( t ) , u ( t )) . (22)Applying, e.g., [2, Theorem 5.4.4] we can immediately deduce the following result, whichclarifies the strengthened form of convergence that we are considering here. This convergenceis strong enough to allow us to pass to the limit in nonlinear functions of u ε : Corollary 3.
Under the same assumptions as in Theorem 2 we have lim ε ↓ Z D f ( x,ξ, u ε ( x, ξ, t )) d γ ε ( x, ξ ) = Z D f ( x, ξ, u ( x, ξ, t )) d γ ( x, ξ ) (23)= 12 Z Ω “ f ( x, − , u − ( x, t )) + f ( x, , u + ( x, t )) ” d λ Ω ( x ) for every t > , where f : D × R → R is an arbitrary continuous function satisfying the quadratic growthcondition | f ( x, ξ, r ) | ≤ A + Br for every ( x, ξ ) ∈ D, r ∈ R for suitable nonnegative constants A, B ∈ R . .9 Structure of the proof Let us briefly explain the structure of the proof of Theorems 1 and 2. This will also clarifythe term Γ-convergence in the title, and highlight the potential of the method for widerapplication.The analogy between (10) and (18) suggests to pass to the limit in these weak formulations,or even better, in their equivalent integrated forms b ε ( u ε ( t ) , v ε ) + Z t a ε ( u ε ( t ) , v ε ) d t = b ( u ε , v ε ) , b ( u ( t ) , v ) + Z t a ( u ( t ) , v ) d t = b ( u , v ) . (24)Applying standard regularization estimates for the solutions to (10) and a weak coercivityproperty of b ε , it is not difficult to prove that u ε ( t ) “weakly” converges to u ( t ) for every t > ρ ε ( t ) = u ε ( t ) γ ε (cid:42) ρ ( t ) = u ( t ) γ weakly- ∗ in M ( D ).The concept of weak convergence of densities that we are using here is thus the same as inTheorem 1, i.e. weak- ∗ convergence of the corresponding measures in M ( D ).In order to pass to the limit in (24) the central property is the following weak-strong convergence principle:For every v ∈ V there exists v ε ∈ V ε with v ε (cid:42) v as ε → u ε (cid:42) u b ε ( u ε , v ε ) → b ( u, v ) and a ε ( u ε , v ε ) → a ( u, v ) . Note that the previous property implies in particular that recovery family v ε converges“strongly” to v , according to the notion considered by Theorem 2, i.e. v ε → v iff v ε (cid:42) v withboth b ε ( v ε , v ε ) → b ( v, v ) and a ε ( v ε , v ε ) → a ( v, v ). Corollary 6 shows that this weak-strongconvergence property can be derived from Γ-convergence in the “weak” topology of the familyof quadratic forms q κε ( u ) := b ε ( u, u ) + κ a ε ( u, u ) to q κ ( u ) := b ( u, u ) + κ a ( u, u ) for κ > . (25)In order to formulate this property in the standard framework of Γ-convergence we will extend a ε and b ε to lower semi-continuous quadratic functionals (possibly assuming the value + ∞ )in the space M ( D ), following the approach of [8, Chap. 11-13]. While the Γ-convergence of b ε is a direct consequence of the weak convergence of γ ε to γ , the convergence of a ε is moresubtle. The convergence of a ε and the structure of the limit depends critically on the choiceof τ ε (defined in (5)): as we show in Section 3.2, the scaling of τ ε in terms of ε is chosenexactly such that the strength of the ‘connection’ between ξ = − ξ = 1 is of order O (1)as ε → b ε = b is a fixed and coercive bilinear form (see, e.g., [4, Chap. 3.9.2]) andcan therefore be considered as the scalar product of the Hilbert space H ε ≡ H . In this casethe embedding of the problems in a bigger topological vector space (the role played by M ( D )in our situation) is no more needed, and one can deal with the weak and strong topologyof H , obtaining the following equivalent characterizations (see e.g. [5, Th. 3.16] and [8, Th.13.6]):1. Pointwise (strong) convergence in H of the solutions of the evolution problems;2. Pointwise convergence in H of the resolvents of the linear operators associated to thebilinear forms a ε ;3. Mosco-convergence in H of the quadratic forms associated to a ε ;4. Γ-convergence in the weak topology of H of the quadratic forms b + κ a ε to b + κ a forevery κ > b ε does depend on ε , Γ-convergence of the extended quadraticforms b ε + κ a ε with respect to the weak- ∗ topology of M ( D ) is thus a natural extension of thelatter condition; Theorem 4 can be interpreted as essentially proving a slightly stronger versionof this property. Starting from this Γ-convergence result, we will derive the convergence ofthe evolution problems by a simple and general argument, which we will present in Section 4. .10 Discussion The result of Theorem 1 is amongst other things a rigorous version of the result of Kramers [12]that was mentioned in the introduction. It shows that the simple reaction-diffusion sys-tem (14) can indeed be viewed as an upscaled version of a diffusion problem in an augmentedphase space; or, equivalently, as an upscaled version of the movement of a Brownian particlein the same augmented phase space.At the same time it generalizes the work of Kramers by adding the spatial dimension,resulting in a limit system which—for this choice of τ ε , see below for more on this choice—captures both reaction and diffusion effects. Measures versus densities.
It is interesting to note the roles of the measures ρ ε , ρ andtheir densities u ε , u with respect to γ ε , γ . The variational formulation of the equations aredone in terms of the densities u ε , u but the limit procedure is better understood in terms of themeasures ρ ε , ρ , since a weak- ∗ convergence is involved. This also allows for a unification of twoproblems with a different structure (a Fokker-Planck equation for u ε and a reaction-diffusionsystem for the couple u − , u + .) Gradient flows.
The weak formulation (10) shows also that a solution u ε can be interpretedas a gradient flow of the quadratic energy a ε ( u, u ) with respect to the L ( D,γ ε ) distance.Another gradient flow structure for the solutions of the same problem could be obtained bya different choice of energy functional and distance: for example, as proved in [11], Fokker-Planck equations like (6) can be interpreted also as the gradient flow of the relative entropyfunctional H ( ρ | γ ε ) := Z D d ρ d γ ε log “ d ρ d γ ε ” d γ ε (26)in the space P ( D ) of probability measures endowed with the so-called L -Wasserstein distance(see e.g. [2]). Other recent work [1] suggests that the Wasserstein setting can be the mostnatural for understanding diffusion as a limit of the motion of Brownian particles, but in thiscase it is not obvious how to interpret the limit system in the framework of gradient flows onprobability measures, and how to obtain it in the limit as ε → The choice of τ ε . In this paper the time scale τ ε is chosen to be equal to ε exp(1 /ε ), and itis a natural question to ask about the limit behaviour for different choices of τ ε . If the scalingis chosen differently—i.e. if τ ε ε − exp( − /ε ) converges to 0 or ∞ —then completely differentlimit systems are obtained: • If τ ε (cid:28) ε exp(1 /ε ), then the reaction is not accelerated sufficiently as ε →
0, and thelimit system will contain only diffusion (i.e. k = 0 in (14)); • If τ ε (cid:29) ε exp(1 /ε ), on the other hand, then the reaction is made faster and faster as ε →
0, resulting in a limit system in which the chemical reaction A (cid:11) B is in continuousequilibrium. Because of this, both A and B have the same concentration u , and u solvesthe diffusion problem ∂ t u = ∆ u, for x ∈ Ω , t > u (0 , x ) = 12 ` u , + ( x ) + u , − ( x ) ´ for x ∈ Ω . Note the instantaneous equilibration of the initial data in this system.While the scaling in terms of ε of τ ε can not be chosen differently without obtainingstructurally different limit systems, there is still a choice in the prefactor. For τ ε := ˜ τ εe /ε with ˜ τ > τ will appear in the definition (15) of k .There is a also a modelling aspect to the choice of τ . In this paper we use no knowledgeabout the value of τ in the diffusion system at finite ε ; the choice τ = τ ε is motivated bythe wish to have a limit system that contains both diffusive and reactive terms. If one hasadditional information about the mobility of the system in the x - and ξ -directions, then thevalue of τ will follow from this. qual rate constants. The assumption of equal depth of the two minima of H correspondsto the assumption (or, depending on one’s point of view, the result) that the rate constant k in (14) is the same for the two reactions A → B and B → A . The general case requires aslightly different choice for H , as follows.Let the original macroscopic equations for the evolution of A and B (in terms of densitiesthat we also denote A and B ) be ∂ t A − ∆ A = k − B − k + A (27a) ∂ t B − ∆ B = k + A − k − B. (27b)Choose a fixed function H ∈ C ∞ ([ − , H (cid:48) ( ±
1) = 0 and H (1) − H ( −
1) =log k − − log k + . We then construct the enthalpy H ε by setting H ε := H + 1 ε H, where H is the same enthalpy function as above. The same proof as for the equal-well casethen gives convergence of the finite- ε problems to (27). Equal diffusion constants.
It is possible to change the setup such that the limiting systemhas different diffusion rate in A and B . We first write equation (6) as ∂ t ρ − div D ε F ε = 0 , where the mobility matrix D ε ∈ R ( d +1) × ( d +1) and the flux F ε are given by D ε = „ I 00 τ ε « and F ε = F ε ( ρ ) = „ ∇ u ∇ ρ + ρ ∇ H « By replacing the identity matrix block I in D ε by a block of the form a ( ξ ) I the x -directionaldiffusion can be modified as a function of ξ . This translates into two different diffusioncoefficients for A and B . The function H . The limit result of Theorem 1 shows that only a small amount ofinformation about the function H propagates into the limit problem: specifically, the localsecond-order structure of H around the wells and around the mountain-pass point.One other aspect of the structure of H is hidden: the fact that we rescaled the ξ variable bya factor of √ τ ε can also be interpreted as a property of H , since the effective distance betweenthe two wells, as measured against the intrinsic distance associated with the Brownian motion,is equal to 2 √ τ ε after rescaling.We also assumed in this paper that H has only ‘half’ wells, in the sense that H is definedon [ − ,
1] instead of R . This was for practical convenience, and one can do essentially thesame analysis for a function H that is defined on R . In this case one will regain a slightlydifferent value of k , namely k = p | H (cid:48)(cid:48) (0) | H (cid:48)(cid:48) (1) / π . (For this reason this is also the valuefound by Kramers [12, equation (17)]). Single particles versus multiple particles, and concentrations versus probabilities.
Thedescription of this paper of the system in terms of a probability measure ρ on D is thedescription of the probability of a single particle. This implies that the limit object ( u − , u + )should be interpreted as the density (with respect to γ ) of a limiting probability measure,again describing a single particle.This is at odds with common continuum modelling philosophy, where the main objects areconcentrations (mass or volume) that represent a large number of particles; in this philosophythe solution ( u − , u + ) of (14) should be viewed as such a concentration, which is to say as theprojection onto x -space of a joint probability distribution of a large number of particles .For the simple reaction A (cid:11) B these two interpretations are actually equivalent. Thisarises from the fact that A → B reaction events in each of the particles are independent ofeach other; therefore the joint distribution of a large number N of particles factorizes intoa product of N copies of the distribution of a single particle. For the case of this paper,therefore, the distinction between these two views is not important. More general reactions.
The remark above implies that the situation will be different forsystems where reaction events cause differences in distributions between the particles, such s the reaction A + B (cid:28) C . This can be recognized as follows: a particle A that has justseparated from a B particle (in a reaction event of the form C → A + B ) has a position thatis highly correlated with the corresponding B particle, while this is not the case for all theother A particles. Therefore the A particles will not have the same distribution. The best onecan hope for is that in the limit of a large number of particles the distribution becomes thesame in some weak way. This is one of the major obstacles in developing a similar connectionas in this paper for more complex reaction equations. One of the main difficulties in the proof of Theorem 1, namely the singular behaviour given bythe concentration of the invariant measure γ ε onto the two lines at ξ = ±
1, can be overcomeby working in the underlying space of (signed or probability) measures in D . This point ofview is introduced in Section 2. Section 3 contains the basic Γ-convergence results (Theorem4) and the proof of Theorem 1 and of Theorem 2. The argument showing the link between Γ-convergence of the quadratic forms a ε , b ε and the convergence of the solutions to the evolutionproblems (see the comments in section 1.9) is presented in Section 4 in a general form, whichcan can be easily applied to other situations. The Kramers-Smoluchowski equation
We first summarize the functional framework introduced above. Let us denote by ( · , · ) ε thescalar product in R d × R defined by( x , y ) ε := x · y + τ ε ξ η, for every x = ( x, ξ ) , y = ( y, η ) ∈ R d × R , (28)with the corresponding norm (cid:107) · (cid:107) ε . We introduced two Hilbert spaces H ε := L ( D, γ ε ) and V ε = W , ( D, γ ε ) , and the bilinear forms b ε ( u, v ) := Z D u v d γ ε for every u, v ∈ H ε , (29) a ε ( u, v ) := Z D ( ∇ x,ξ u, ∇ x,ξ v ) ε d γ ε for every u, v ∈ V ε , (30)with which (7) has the variational formulation b ε ( ∂ t u ε , v ) + a ε ( u ε , v ) = 0 for every v ∈ V ε , t > u ε (0 , · ) = u ε . (31)The main technical difficulty in studying the limit behaviour of (31) as ε ↓ ε -dependence of the functional spaces H ε , V ε . Since for our approach it is crucial to workin a fixed ambient space, we embed the solutions of (31) in the space of finite Borel measures M ( D ) by associating to u ε the measure ρ ε := u ε γ ε . We thus introduce the quadratic forms b ε ( ρ ) := b ε ( u, u ) if ρ (cid:28) γ ε and u = d ρ d γ ε ∈ H ε , (32) a ε ( ρ ) := a ε ( u, u ) if ρ (cid:28) γ ε and u = d ρ d γ ε ∈ V ε , (33)trivially extended to + ∞ when ρ is not absolutely continuous with respect to γ ε or its density u does not belong to H ε or V ε respectively. Denoting by Dom ( a ε ) and Dom ( b ε ) their properdomains, we still denote by a ε ( · , · ) and b ε ( · , · ) the corresponding bilinear forms defined on Dom ( a ε ) and Dom ( b ε ) respectively. Setting ρ ε := u ε γ ε , σ := vγ ε , (31) is equivalent to theintegrated form b ε ( ρ ε ( t ) , σ ) + Z t a ε ( ρ ε ( r ) , σ ) d r = b ε ( ρ ε , σ ) for every σ ∈ Dom ( a ε ) . (34) e also recall the standard estimates12 b ε ( ρ ε ( t )) + Z t a ε ( ρ ε ( r )) d r = 12 b ε ( ρ ε ) for every t ≥ , (35) t a ε ( ρ ε ( t )) + 2 Z t r b ε ( ∂ t ρ ε ( r )) d r = Z t a ε ( ρ ε ( r )) d r for every t ≥ , (36)12 b ε ( ρ ε ( t )) + t a ε ( ρ ε ( t )) + t b ε ( ∂ t ρ ε ( t )) ≤ b ε ( ρ ε ) for every t > . (37)Although versions of these expressions appear in various places, we were unable to find areference that completely suits our purposes. We therefore briefly describe their proof, andwe use the more conventional formulation in terms of the bilinear forms a ε and b ε and spaces H ε and V ε ; note that b ε is an inner product for H ε , and b ε + a ε is an inner product for V ε .When u is sufficiently smooth, standard results (e.g. [6, Chapter VII]) provide the exis-tence of a solution u ε ∈ C ([0 , ∞ ); V ε ) ∩ C ∞ ((0 , ∞ ); V ε ), such that the functions t (cid:55)→ a ε ( u ε ( t ))and t (cid:55)→ b ε ( ∂ t u ε ( t )) are non-increasing; in addition, the solution operator (semigroup) S t is acontraction in H ε . For this case all three expressions can be proved by differentiation.In order to extend them to all u ε ∈ H ε , we note that for fixed t > H ε given by (the square roots of) u ε (cid:55)→ b ε ( u ε ) and u ε (cid:55)→ b ε ( S t u ε ) + Z t a ε ( S r u ε ) d r (38)are identical by (35) on a H ε -dense subset. If we approximate a general u ε ∈ H ε by smooth u ε,n , then the sequence u ε,n is a Cauchy sequence with respect to both norms; by copyingthe proof of completeness of the space L (0 , ∞ ; V ε ) (see e.g. [6, Th. IV.8]) it follows that theintegral in (38) converges. This allows us to pass to the limit in (35). The argument is similarfor (37), when one writes the sum of (35) and (36) as12 b ε ( u ε ( t )) + ta ε ( u ε ( t )) + 2 Z t rb ε ( ∂ t u ε ( r )) d r = 12 b ε ( u ε ) . (39)Finally, (37) follows by (39) since r (cid:55)→ b ε ( ∂ t u ε ( r )) is non-increasing. The reaction-diffusion limit
We now adopt the same point of view to formulate the limit reaction-diffusion system in thesetting of measures. Recall that for u ∈ H := L ( D, γ ) we set u ± ( x ) := u ( x, ± V := ˘ u ∈ H : u ± ∈ W , (Ω) ¯ , and the bilinear forms b ( u, v ) = 12 Z Ω “ u + v + + u − v − ” d λ Ω , (40) a ( u, v ) := 12 Z Ω “ ∇ x u + ∇ x v + + ∇ x u − ∇ x v − + k ` u + − u − ´ ( v + − v − ) ” d λ Ω . (41)As before we now extend these definitions to arbitrary measures by b ( ρ ) := b ( u, u ) if ρ (cid:28) γ and u = d ρ d γ ∈ H, (42) a ( ρ ) := a ( u, u ) if ρ (cid:28) γ and u = d ρ d γ ∈ V , (43)with corresponding bilinear forms b ( · , · ) and a ( · , · ); problem (14a,b,c) can be reformulated as b ( ∂ t ρ ( t ) , σ ) + a ( ρ ( t ) , σ ) = 0 for every t > σ ∈ Dom ( a ) , or in the integral form b ( ρ ( t ) , σ ) + Z t a ( ρ ( r ) , σ ) d r = b ( ρ , σ ) for every σ ∈ Dom ( a ) . (44)Since both problems (34) and (44) are embedded in the same measure space M ( D ), wecan study the convergence of the solution ρ ε of (34) as ε ↓ Γ -convergence result for the quadratic forms a ε , b ε The aim of this section is to prove the following Γ-convergence result:
Theorem 4. If ρ ε (cid:42) ρ as ε ↓ in M ( D ) then lim inf ε ↓ a ε ( ρ ε ) ≥ a ( ρ ) , lim inf ε ↓ b ε ( ρ ε ) ≥ b ( ρ ) . (45) For every ρ ∈ M ( D ) such that a ( ρ ) + b ( ρ ) < + ∞ there exists a family ρ ε ∈ M ( D ) weakly- ∗ converging to ρ such that lim ε ↓ a ε ( ρ ε ) = a ( ρ ) , lim ε ↓ b ε ( ρ ε ) = b ( ρ ) . (46)Note that M ( D ) endowed with the weak- ∗ topology is the dual of a separable Banachspace, and therefore the sequential definition of Γ-convergence coincides with the topologicaldefinition [8, Proposition 8.1 and Theorem 8.10]; consequently Theorem 4 implies the Γ-convergence of the families a ε and b ε . Theorem 4 actually states a stronger result, since therecovery sequence can be chosen to be the same for a ε and b ε . This joint Γ-convergence ofthe families a ε and b ε is nearly equivalent with Γ-convergence of combined quadratic forms: Lemma 5.
Theorem 4 implies the Γ( M ( D )) -convergence of q κε ( ρ ) := b ε ( ρ ) + κ a ε ( ρ ) to q κ ( ρ ) := b ( ρ ) + κ a ( ρ ) (47) for each κ > .Conversely, if we assume (47) , then (46) holds, and (45) follows under the additionalassumption lim sup ε ↓ a ε ( ρ ε ) + b ε ( ρ ε ) = C < + ∞ . (48) Proof.
The first part of the Lemma is immediate. For the second part, suppose that ρ ε (cid:42) ρ and satisfies (48); the Γ-liminf inequality for q κε yieldslim inf ε ↓ b ε ( ρ ε ) ≥ lim inf ε ↓ q kε ( ρ ε ) − Cκ ≥ q κ ( ρ ) − Cκ = b ( ρ ) + κ ` a ( ρ ) − C ´ for every κ > , and therefore the second inequality of (45) follows by letting κ ↓
0. A similar argument yieldsthe first inequality of (45).Concerning (46), Γ-convergence of q ε to q yields a recovery family ρ ε (cid:42) ρ such thatlim ε ↓ a ε ( ρ ε ) + b ε ( ρ ε ) = a ( ρ ) + b ( ρ ) < + ∞ ;In particular a ε ( ρ ε )+ b ε ( ρ ε ) is uniformly bounded, so that (45) yields the separate convergence(46).One of the most useful consequences of (47) is contained in the next result (see e.g. [16,Lemma 3.6]). Corollary 6 (Weak-strong convergence) . Assume that (47) holds for every κ > and let ρ ε , σ ε ∈ M ( D ) be two families weakly converging to ρ, σ as ε ↓ and satisfying the uniformbound (48) , i.e. lim sup ε ↓ a ε ( ρ ε ) + b ε ( ρ ε ) < + ∞ , lim sup ε ↓ a ε ( σ ε ) + b ε ( σ ε ) < + ∞ , (49) so that ρ, σ belong to the domains of the bilinear form a and b . We have lim ε ↓ a ε ( σ ε ) = a ( σ ) = ⇒ lim ε ↓ a ε ( ρ ε , σ ε ) = a ( ρ, σ ) (50)lim ε ↓ b ε ( σ ε ) = b ( σ ) = ⇒ lim ε ↓ b ε ( ρ ε , σ ε ) = b ( ρ, σ ) . (51) roof. We reproduce here the proof of [16] in the case of the quadratic forms a ε (50). Notethat by (49) and Lemma 5 we can assume that ρ ε and σ ε satisfy (45). For every positivescalar r > a ε ( ρ ε , σ ε ) = 2 a ε ( r ρ ε , r − σ ε ) = a ε ( rρ ε + r − σ ε ) − r a ε ( ρ ε ) − r − a ε ( σ ε ) . Taking the inferior limit as ε ↓ A := lim sup ε ↓ a ε ( ρ ε )lim inf ε ↓ a ε ( ρ ε , σ ε ) ≥ a ( rρ + r − σ ) − r A − r − a ( σ ) = 2 a ( ρ, σ ) + r ` a ( ρ ) − A ´ . Since r > A is finite by (49) we obtain lim inf ε ↓ a ε ( ρ ε , σ ε ) ≥ a ( ρ, σ ) andinverting the sign of σ we get (50).We split the proof of Theorem 4 in various steps. Ω × {− , } . Lemma 7. If ρ ε = u ε γ ε satisfies the uniform bound a ε ( ρ ε ) ≤ C < + ∞ for every ε > , thenfor every δ ∈ (0 , ∂ ξ u ε → in L (Ω × ω δ ) , as ε → , (52) where ω δ := ( − , − δ ) ∪ ( δ, .Proof. We observe that τ ε Z D ( ∂ ξ u ε ) d γ ε ≤ a ε ( ρ ε ) ≤ C < ∞ . If h δ = sup ξ ∈ ω δ H ( ξ ) <
1, then inf ξ ∈ ω δ e − H ( ξ ) /ε = e − h δ /ε , and we find Z Ω × ω δ ( ∂ ξ u ε ) d x d ξ ≤ C Z ε τ ε e hδε = C Z ε ε e hδ − ε . Taking the limit as ε → Lemma 8 (Convergence of traces) . Let us suppose that ρ ε = u ε γ ε (cid:42) ρ = uγ with a ε ( ρ ε ) ≤ C < + ∞ and let u ± ε ( x ) be the traces of u ε at ξ = ± . Then as ε ↓ u ± ε → u ± strongly in L (Ω) , (53) where u ± are the functions given by (13) .Proof. Let us consider, e.g., the case of u − ε . Let us fix δ ∈ (0 , W , ( − , − δ ) we know thatlim ε ↓ Z Ω ω ε ( x ) d L d = 0 where ω ε ( x ) := sup − ≤ ξ ≤− δ | u ε ( x, ξ ) − u − ε ( x ) | ≤ δ Z − δ − | ∂ ξ u ε ( x, ξ ) | d ξ. (54)Let us fix a function φ ∈ C (Ω) and a function ψ ∈ C [ − ,
1] with 0 ≤ ψ ≤ ψ ( −
1) = 1,supp ψ ⊂ [ − , − δ ]; we set J ε := Z − ψ ( ξ ) d˜ γ ε ( ξ ) , ˜ u ε ( x ) := J − ε Z − u ε ( x, ξ ) ψ ( ξ ) d ˜ γ ε ( ξ ) , where ˜ γ ε is the measure defined in (3). Note thatlim ε → J ε = (cid:104) ψ, γ (cid:105) = 12 ψ ( −
1) + 12 ψ (1) = 12 . Since ρ ε weakly converge to ρ we know thatlim ε ↓ Z Ω φ ( x )˜ u ε ( x ) d λ Ω = lim ε ↓ J − ε Z Ω φ ( x ) ψ ( ξ ) u ε ( x, ξ ) d γ ε ( x, ξ ) = Z Ω φ ( x ) u − ( x ) d λ Ω o that ˜ u ε converges to u − in the duality with bounded continuous functions. On the otherhand, Z Ω |∇ x ˜ u ε ( x ) | d λ Ω ≤ J − ε Z Ω Z − |∇ x u ε ( x, ξ ) | ψ ( ξ ) d˜ γ ( ξ ) d λ Ω ( x ) ≤ J − ε a ε ( ρ ε ) ≤ C so that ˜ u ε → u − in L (Ω) by Rellich compactness theorem.On the other hand, thanks to (54), we havelim ε ↓ Z Ω ˛˛˛ u − ε ( x ) − ˜ u ε ( x ) ˛˛˛ d λ Ω ( x ) = lim ε ↓ J − ε Z Ω ˛˛˛ Z − ψ ( ξ ) ` u ε ( x, ξ ) − u − ( x ) ´ d˜ γ ε ( ξ ) ˛˛˛ d λ Ω ( x ) ≤ lim ε ↓ Z D ψ ( ξ ) ω ε ( x ) d γ ε ( x, ξ ) = 0 , which yields (53). Remark.
A completely analogous argument shows that if ρ ε satisfies a W , ( D ; γ ε )-uniformbound Z D (cid:107)∇ x,ξ u ε (cid:107) ε d γ ε ( x, ξ ) ≤ C < + ∞ (55)instead of a ε ( ρ ε ) ≤ C , then u ± ε → u ± in L (Ω). Given ( ϕ − , ϕ + ) ∈ R let us set K ε ( ϕ − , ϕ + ) := min n τ ε Z − ` ϕ (cid:48) ( ξ ) ´ d ˜ γ ε : ϕ ∈ W , ( − , , ϕ ( ±
1) = ϕ ± o (56)It is immediate to check that K ε is a quadratic form depending only on ϕ + − ϕ − , i.e. K ε ( ϕ − , ϕ + ) = k ε ( ϕ + − ϕ − ) , k ε = K ε ( − / , / . (57)We call T ε ( ϕ − , ϕ + ) the solution of the minimum problem (56): it admits the simple repre-sentation T ε ( ϕ − , ϕ + ) = 12 ( ϕ − + ϕ + ) + ( ϕ + − ϕ − ) φ ε (58)where φ ε = T ε ( − / , / Q ε ( ϕ − , ϕ + ) := Z − ` T ε ( ϕ − , ϕ + ) ´ d˜ γ ε = 12 ` ( ϕ − ) + ( ϕ + ) ´ + ( q ε − )( ϕ + − ϕ − ) (59)where q ε := Z − | φ ε ( ξ ) | d˜ γ ε ( ξ ) = Q ε ( − / , / . (60) Lemma 9.
We have lim ε ↓ k ε = k p − H (cid:48)(cid:48) (0) H (cid:48)(cid:48) (1)2 π , (61) and lim ε ↓ q ε = 14 so that lim ε ↓ Q ε ( ϕ − , ϕ + ) = 12 ( ϕ − ) + 12 ( ϕ + ) . (62) Proof. φ ε solves the Euler equation ` e − H ( ξ ) /ε φ (cid:48) ε ( ξ ) ´ (cid:48) = 0 on ( − , , φ ε ( ±
1) = ± . (63)We can compute an explicit solution of (63) by integration: φ (cid:48) ε ( ξ ) = Ce H ( ξ ) /ε , φ ε ( ξ ) = C (cid:48) + C Z ξ e H ( η ) /ε d η. efine I ε := R − e H ( ξ ) /ε d ξ . The boundary conditions for ξ = ± C (cid:48) = 0 , C Z − e H ( ξ ) /ε dξ = CI ε = 1It follows that φ ε ( ξ ) = I − ε Z ξ e H ( η ) /ε d η, and k ε = τ ε I − ε Z − e H ( ξ ) /ε d˜ γ ε ( ξ ) = τ ε Z − I − ε . We compute, using Laplace’s method: I ε = s πε | H (cid:48)(cid:48) (0) | e /ε (1 + o (1)) and Z ε = s πεH (cid:48)(cid:48) (1) (1 + o (1)) , as ε → , thus obtaining (61). Since φ (cid:48) ε = I − ε e H/ε → δ in D (cid:48) ( − , H is even, we have φ ε ( ξ ) = I − ε Z ξ e H ( η ) /ε d η →
12 sign( ξ )uniformly on each compact subset of [ − ,
1] not containing 0. Since the range of φ ε belongsto [ − / , /
2] and ˜ γ ε (cid:42) δ − + δ +1 we obtain (62). The second limit of (45) follows by general lower semicontinuity results on integral functionalsof measures, see e.g. [2, Lemma 9.4.3].Concerning the first “lim inf” inequality, we split the the quadratic form a ε in the sum oftwo parts, a ε ( ρ ε ) := Z D |∇ x u ε ( x, ξ ) | d γ ε ( x, ξ ) , a ε ( ρ ε ) := τ ε Z D ( ∂ ξ u ε ) d γ ε ( x, ξ ) . (64)We choose a smooth cutoff function η − : [ − , → [0 ,
1] such that η − ( −
1) = 1 and supp( η − ) ⊂ [ − , − /
2] and the symmetric one η + ( ξ ) := η ( − ξ ). We also set u − ε ( x ) := Z − η − ( ξ ) u ε ( x, ξ ) d˜ γ ε ( ξ ) , u + ε ( x ) := Z − η + ( ξ ) u ε ( x, ξ ) d˜ γ ε ( ξ ) , (65)and it is easy to check that u ± ε (cid:42) u ± in D (cid:48) (Ω) . (66)We also set θ ε := R − η + ( ξ ) d˜ γ ε ( ξ ) ` = R − η − ( ξ ) d˜ γ ε ( ξ ) ´ , observing that θ ε → /
2. We thenhave by Jensen inequality a ε ( ρ ε ) ≥ Z Ω Z − ( η − ( ξ ) + η + ( ξ )) |∇ x u ε | d˜ γ ε ( ξ ) d λ Ω ≥ θ − ε Z Ω |∇ u − ε | + |∇ u + ε | d λ Ω and, passing to the limit,lim inf ε ↓ a ε ( ρ ε ) ≥ Z Ω |∇ u − | + |∇ u + | d λ Ω Let us now consider the behaviour of a ε : applying (56) and (57) we get a ε ( ρ ε ) = Z Ω „ τ ε Z − ( ∂ ξ u ε ( x, ξ )) d˜ γ ε ( ξ ) « d λ Ω ≥ Z Ω k ε ( u − ε ( x ) − u + ε ( x )) d λ Ω o that by (61) and (53) we obtainlim inf ε ↓ a ε ( ρ ε ) ≥ k Z Ω ` u − ( x ) − u + ( x ) ´ d λ Ω . (67)In order to prove the “lim sup” inequality (46) we fix ρ = uγ with u in the domain of thequadratic forms a and b so that u ± = u ( · , ±
1) belong to W , (Ω), and we set ρ ε = u ε γ ε where u ε ( x, · ) = T ε ( u − ( x ) , u + ( x )) as in (58). We easily have by (62) and the Lebesgue dominatedconvergence theoremlim ε ↓ b ε ( ρ ε ) = lim ε ↓ Z Ω Q ε ( u − ( x ) , u + ( x )) d λ Ω = Z Ω “ | u − ( x ) | + 12 | u + ( x ) | ” d λ Ω = b ( ρ ) . Similarly, since for every j = 1 , · · · , d and almost every x ∈ Ω ∂ x j u ε ( x, ξ ) = T ( ∂ x j u − ( x ) , ∂ x j u + ) , we havelim ε ↓ a ε ( ρ ε ) = lim ε ↓ Z Ω “ d X j =1 Q ε ` ∂ x j u − ( x ) , ∂ x j u + ( x ) ´ + K ε ` u − ( x ) , u + ( x ) ´” d λ Ω == Z Ω “ |∇ u − ( x ) | + 12 |∇ u + ( x ) | + k ` u − ( x ) − u + ( x ) ´” d λ Ω = a ( ρ ) . Γ -convergence to convergence of the evolu-tion problems: proof of Theorems 1 and 2. Having at our disposal the Γ-convergence result of Theorem 4 and its Corollary 6 it is notdifficult to pass to the limit in the integrated equation (34).Let us first notice that the quadratic forms b ε satisfy a uniform coercivity condition: Lemma 10 (Uniform coercivity of b ε ) . Every family of measures ρ ε ∈ M ( D ) , ε > satisfying lim sup ε> b ε ( ρ ε ) < + ∞ (68) is bounded in M ( D ) and admits a weakly- ∗ converging subsequence.Proof. It follows immediately by the fact that γ ε is a probability measure and therefore | ρ ε | ( D ) ≤ “ b ε ( ρ ε ) ” / . (68) thus implies that the total mass of ρ ε is uniformly bounded and we can apply the relativeweak- ∗ compactness of bounded sets in dual Banach spaces.The proof of Theorems 1 and 2 is a consequence of the following general result: Theorem 11 (Convergence of evolution problems) . Let us consider weakly- ∗ lower-semiconti-nuous, nonnegative and extended-valued quadratic forms a ε , b ε , a , b defined on M ( D ) and letus suppose that Non degeneracy of the limit forms: b is non degenerate (i.e. b ( ρ ) = 0 ⇒ ρ = 0 )and Dom ( a ) is dense in Dom ( b ) with respect to the norm-convergence induced by b . Uniform coercivity: b ε satisfy the coercivity property stated in the previous Lemma10. Joint Γ -convergence: q κε := b ε + κ a ε satisfy the joint Γ -convergence property (47)Γ ` M ( D ) ´ - lim ε ↓ q κε = q κ = b + κ a for every κ > . (69) et ρ ε ( t ) , t ≥ , be the solution of the evolution problem (34) starting from ρ ε ∈ Dom ( b ε ) .If ρ ε (cid:42) ρ in M ( D ) as ε ↓ with lim sup ε ↓ b ε ( ρ ε ) < + ∞ (70) then ρ ε ( t ) (cid:42) ρ ( t ) in M ( D ) as ε ↓ for every t > and ρ ( t ) is the solution of the limitevolution problem (44) .If moreover lim ε ↓ b ε ( ρ ε ) = b ( ρ ) then lim ε ↓ b ε ( ρ ε ( t )) = b ( ρ ( t )) , lim ε ↓ a ε ( ρ ε ( t )) = a ( ρ ( t )) for every t > . (71) Proof.
Let us first note that by (35) and the coercivity property of b ε the mass of ρ ε ( t ) isbounded uniformly in t . Moreover, (37) and the coercivity property show that ∂ t ρ ε is a finitemeasure whose total mass is uniformly bounded in each bounded interval [ t , t ] ⊂ (0 , + ∞ ).By the Arzela-Ascoli theorem we can extract a subsequence ρ ε n such that ρ ε n ( t ) (cid:42) ρ ( t ) forevery t ≥
0. The estimates (37) and (45) show that for every t > ρ ( t ) belongs to the domainof the quadratic forms a and b , and satisfies a similar estimate12 b ( ρ ( t )) + t a ( ρ ( t )) + t b ( ∂ t ρ ( t )) ≤
12 lim inf ε ↓ b ( ρ ε ) < + ∞ . (72)Let σ ∈ M ( D ) be an arbitrary element of the domains of a and b ; by (47) we can finda family σ ε (actually a family σ ε n , but we suppress the subscript n ) weakly converging to σ such that (46) holds. By (34) we have b ε ( ρ ε ( t ) , σ ε ) + Z t a ε ( ρ ε ( r ) , σ ε ) d r = b ε ( ρ ε , σ ε ) (73)and (37) with the Schwarz inequality yields the uniform bound ˛˛ a ε ( ρ ε ( t ) , σ ε ) ˛˛ ≤ t − / b ε ( ρ ε ) / a ε ( σ ε ) / ≤ Ct − / where C is independent of ε ; we can therefore pass to the limit in (73) by Corollary 6 to find b ( ρ ( t ) , σ ) + Z t a ε ( ρ ( r ) , σ ) d r = b ( ρ , σ ) , so that ρ is a solution of the limit equation. Since the limit is uniquely identified by thenon-degeneracy and density condition 1), we conclude that the whole family ρ ε converges to ρ as ε ↓
0. In particular ρ satisfies the identity12 b ( ρ ( t )) + Z t a ( ρ ( r )) d r = 12 b ( ρ ) for every t ≥ . (74)This concludes the proof of (70) (and of Theorem 1).In order to prove (71) (and Theorem 2) we note that by (35) and (74) we easily getlim sup ε ↓ b ε ( ρ ε ( t )) + Z t a ε ( ρ ε ( r )) d r ≤ b ( ρ ( t )) + Z t a ( ρ ( r )) d r. The lower-semicontinuity property (45) and Fatou’s Lemma yieldlim ε ↓ b ε ( ρ ε ( t )) = b ( ρ ( t )) , lim ε ↓ Z t a ε ( ρ ε ( r )) d r = Z t a ( ρ ( r )) d r for every t ≥ . (75)Applying the same argument to (36) and its “ ε = 0” analogue we conclude that a ε ( ρ ε ( t )) → a ( ρ ( t )) for every t > Remark (More general ambient spaces) . The particular structure of M ( D ) did not play anyrole in the previous argument, so that the validity of the above result can be easily extendedto general topological vector spaces (e.g. dual of separable Banach spaces with their weak- ∗ topology), once the coercivity condition of Lemma 10 is satisfied. eferences [1] S. Adams, N. Dirr, M. A. Peletier, and J. Zimmer. Foundation of the Wassersteingradient-flow formulation of diffusion: A large-deviation approach. In preparation.[2] L. Ambrosio, N. Gigli, and G. Savar´e. Gradient Flows in Metric Spaces and in the Spaceof Probability Measures . Lectures in mathematics ETH Z¨urich. Birkh¨auser, 2005.[3] R. Aris.
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