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Fast Diffusion Limit for Reaction-Diffusion Systems with Stochastic Neumann Boundary Conditions

We consider a class of reaction-diffusion equations with a stochastic perturbation on the boundary. We show that in the limit of fast diffusion, one can rigorously approximate solutions of the system of PDEs with stochastic Neumann boundary conditions by the solution of a suitable stochastic/deterministic differential equation for the average concentration that involves reactions only. An interesting effect occurs, if the noise on the boundary does not change the averaging concentration, but is sufficiently large. Then surprising additional effective reaction terms appear. We focus on systems with polynomial nonlinearities only and give applications to the two dimensional nonlinear heat equation and the cubic auto-catalytic reaction between two chemicals.


Introduction
Stochastic partial differential equations (SPDEs) appear naturally as models for dynamical systems with respect to random influences. Sometimes in a complex physical system the noise has an impact not only on the bulk of the system but on its physical boundary, too. This happens for instance in heat transfer in a solid in contact with a fluid [11], chemical reactor theory [12], colloid and interface chemistry [19], and the air-sea interactions on the ocean surface [17]. Let G be a bounded sufficiently smooth domain in R d for d ≥ 1, which has a smooth boundary ∂G. We consider the following system of stochastic reactiondiffusion equations for n species with respect to random Neumann boundary where A is the diffusion term, the reaction terms F i (u 1 , u 2 , ...., u n ) are polynomials of degree m i , W i are independent Q-Wiener process in L 2 (∂G), and ∂u ∂ν is the normal derivative of u on ∂G.
The assumption of independence is mainly for convenience of presentation, as now some terms cancel and the technicalities are less involved. Sowers [15] investigated multidimensional stochastic reaction diffusion equation with Neumann boundary conditions and he showed that there is a unique solution. Da Prato and Zabczyk [8,9] discussed the difference between the problems with Dirichlet and Neumann boundary noises, while [1,3] study random Dirichlet boundary conditions. Other results are [13,2].
An very interesting result is by Schnaubelt and Veraar [18], where regularity of solutions is studied. Furthermore, mild and weak solutions are shown to coincide.
Recently, Cerrai and Freidlin [4] considered a class of stochastic reactiondiffusion equations with Neumann boundary noise. Also, they showed that when the diffusion rate is much larger than the rate of reaction, it is possible to replace the SPDE by a suitable one-dimensional stochastic differential equation. But their result only allowed for weak convergence of the approximation without any order of the error.
Our aim is to establish rigorously error bounds results for the fast-diffusion limit for the general class of PDEs with stochastic Neumann boundary conditions given by (1). The error estimates are performed in an L p -space setting, as we cannot expect solutions to (1) to be smooth. Especially, at the forced boundary the solution u is expected to be even unbounded, although it is smoother inside the domain. See [18] or for Dirichlet boundary [1].
We consider two cases. The second on is the relatively simple limit, where the fast diffusion just disappears in the limit, while in the first case large noise changes the limiting reaction equation. The reason for large noise might be that both diffusion and noise are enhanced by stirring.
First case: If the noise does not change the average (W c = 0) but is sufficiently large (σ ε = ε −1 ), then the solutions of Equation (1) are well approximated by u(t, x) ≃ b(t) + Z s (t, x) + error, where b(t) ∈ R n represents the average concentration of the components of u given in general formulation as a solution of for some polynomial G of degree less than or equal m − 2 depending on the structure of the noise. The stochastic perturbation Z s (t, x) is defined later in (27). It is an ε-dependent fast Ornstein-Uhlenbeck process (OU-process) corresponding to white noise in the limit ε → 0. The index c denotes the average (i.e., v c = |G| −1 G vdx which is the projection onto the constants). The ODE ∂ t b(t) = F (b) is the expected result, but due to noise an additional term of noise induced effective reactions appears. We illustrate our results using a relatively simple auto-catalytic reaction. For the result presented we always need a square which averages to a constant in the limit ε → 0. This is mainly, because we assumed independent noise terms for each species. In contrast, if the noise terms are dependent, then any reaction term could lead to an additional effective reaction term in the limit.
Second case: If W c = 0 and σ ε = 1, then the solution of Equation (1) are well approximated by and b is the solution of stochastic ordinary differential equation for some Wiener processβ in R n , which is essentially the projection of W onto the dominant constant modes, i.e. the direct impact of the noise on the average. This is the somewhat expected result, where the reaction-diffusion equation under fast diffusion is well approximated by the reaction ODE.
As an application of our results, we give some examples from physics (nonlinear heat equation) and from chemistry (cubic auto-catalytic reaction between two chemicals according to the rule A + B → 2B). To illustrate our results let us focus for a moment on the relatively simple two dimensional nonlinear heat equation (also called Ginzburg-Landau or Allen-Cahn), which is partly covered by the setting of [4], too.
For the first case we suppose W c = 0 and σ ε = ε −1 , and our main Theorem 17 states that the solution of (6) is well approximated by (2) and b is the solution where C α,λ is a constant depending on the noise intensity parameters α i,k and the eigenvalues of the operator ∆. For the second case W c = 0 and σ ε = 1 our main Theorem 20 states that the solution of (6) is of the form (4) and b is the solution of where B is a R-valued standard Brownian motion.
The main novelties of this paper are on one hand the explicit error estimate in terms of high moments of the error, as usually only weak convergence is treated (see e.g. [4]), and on the other hand the observation that large massconservative noise has the potential to change effective reaction equations in the limit of large diffusion.
The paper is organized as follows. Our assumptions and some definitions are given in the next section. In Section 3 we derive the fast-diffusion limit with error terms and present the main theorem. Section 4 gives bounds for high non-dominant modes, while Section 5 provides averaging results over the fast OU-process. In Section 6, we give the proof of the approximation Theorem I and some examples from physics and chemistry as applications of our results. Finally, we prove the approximation Theorem II and apply this result to nonlinear heat equation and cubic auto-catalytic reaction between two chemicals.

Definition and Assumptions
This section states the precise setting for (1) and summarizes all assumptions necessary for our results. For the analysis we work in the separable Hilbert space L 2 (G) of square integrable functions, where G ⊂ R d is a bounded domain with sufficiently smooth boundary ∂G (e.g. Lipschitz), equipped with scalar product ·, · and norm · .
Definition 1 Define for i = 1, 2, ...., n and diffusion constants d i > 0 where ∂ ν u is the normal derivative of u on ∂G.
Let {g k } ∞ k=1 be an orthonormal basis of eigenfunctions of A i in L 2 (G). It is obviously the same basis for all i with corresponding eigenvalues {d i λ k } ∞ k=0 depending on i (cf. Courant and Hilbert [5]). Also, let {e k } n k=1 be the standard orthonormal basis of R n . Hence, {g k e i } for k ∈ N 0 and i = {1, .., n}, is an orthonormal basis of A in L 2 (G) n such that A (g k e i ) = −d i λ k g k e i .

Assumption 2
We assume that for all k ∈ N This is true in R 2 for instance on squares, hexagons, and triangles with γ 1 = 0, while the worst case is γ 1 = (d − 1)/2 realized for balls and spheres. See [6]. This condition might be relaxed, but we focused in examples mainly on cases with γ 1 = 0.
Define N := ker A = span{e 1 g 0 , ...., e n g 0 }, 2 is a constant and λ 0 = 0. Define S = N ⊥ to be the orthogonal complement of N in L 2 (G) n . Denote by P c u = 1 |G| G udx the projection onto N and define P s u := (I − P c )u for the projection onto the orthogonal complement, where I is the identity operator on L 2 (G) n . We define L p n := [L p (G)] n . The operator A given by Definition 1 generates an analytic semigroup {e tA } t≥0 (cf. Dan Henry [10] or Pazy [16]), on L p n for all p ≥ 2. It has the following property: There is an ω > 0 such that for all t > 0 and all u ∈ L p n e tA P s u L p n ≤ M e −ωt P s u L p n , where ω depends in general on d i . Moreover, we obtain Also, we suppose Assumption 3 There is a constant M > 0 such that for all t > 0 and u ∈ L mp n e tA u L mp The previous assumption is needed for the existence of the solutions and global bounds. Equation (10) follows the Sobolev-embedding of W α,p into L mp . The main assumption is that the coefficient is between 0 and 1.
Immediate conclusion of Assumption 3 and Equation (8) is where for simplicity we denote different constants ω, M by the same name.
For the noise we suppose: Assumption 4 Let W = (W 1 , ...., W n ) be a collection of n independent Wiener process on an abstract probability space (Ω, F , P) with a bounded covariance operator Q i : where (α i,k ) k∈N0 is a bounded sequence of real numbers and (f k ) k∈N0 be any orthonormal basis on L 2 (∂G) with f 0 ≡Constant. For t ≥ 0 we can write W i (t) (cf. Da Prato and Zabczyk [7]) as where (β i,k ) k∈N0 are independent, standard Brownian motions in R. Also, we assume by using the orthonormal basis g k of A i in L 2 (G) that for some small where the covariance q i,j k,ℓ is defined by For the nonlinearity we assume Assumption 5 The nonlinearity F is a polynomial of at most degree m. Thus for all p ≥ 1 it is bounded by where m = max(m 1 , ....., m n ) and the m i are the degrees of the polynomials F i .
The following assumption ensures, that the noise is mass-conserving and that various series converge. This is used in Case 1 only.
The following two assumptions are used in the two cases separately. They are usually lemmas that follows directly from the fact that F is a polynomial. Note that T 1 in general depends on the initial condition b(0).
Assumption 9 Let b(t) in N be the solution of (5). Suppose there is a stopping time T 1 ≤ T 0 and C > 0, such that for sufficiently large ζ ≫ 1 and for δ > 0 and κ ∈ (0, 1 m+1 ) We remark that ζ depends mainly on T 0 and κ (cf. Section 8).
For our result we rely on a cut off argument. We consider only solutions u that are not too large, as given by the next definition.
Definition 10 For a mild solution u of (1) we define for κ ∈ (0, 1 m+1 ) the stopping time τ * as We give error estimates in terms of the following O-notation.
Definition 11 For a real-valued family of processes {X ε (t)} t≥0 we say that X ε is of order f ε , i.e.
We use also the analogous notation for time-independent random variables.

Random boundary conditions
Definition 13 (Neumann map) The Neumann map D : With a slight abuse of notation, we also denote by D the extension from L 2

Definition 14
Define the stochastic convolution Z(t) as The next lemma expands the stochastic convolution Z as a Fourier series.
Lemma 15 Under Assumption 4 let Z be the stochastic convolution defined in (22), then (withW i,j defined in (15)) Proof. Writing Z(t) in Fourier expansion yields

Using Equation (22)
where we used Gauss-Green formula and Definition 13. Hence It is easy to check, that this series converges in L 2 n .

Limiting equation and main theorem
In this section we derive formally the limiting equation for (1) and we state without proof the main theorem of this paper. First, let us define the mild solution of Equation (1) according to [8,9] as follows Definition 16 For any fixed ε > 0, we call a L p n -valued stochastic process u a mild solution of (1) in L p n if for all t > 0 up to a positive stopping time Because we are working with a locally Lipschitz nonlinearity, under Assumption 3, the existence and uniqueness of solutions is standard, once Z is sufficiently regular. See e.g. [7] and [9]. We can rewrite Equation (24) by using Equation (23) as withW i,j defined in (15). Now, let us discuss two cases depending on σ ε and α i,0 for i = 1, ..., n.
4.1 First case: σ ε = ε −1 and α i,0 = 0 for i = 1, .., n In this case Equation (25) takes the form where with and In order to derive the limiting equation, we split the solution u into with a ∈ N and ψ ∈ S. Plugging (30) into (26) and projecting everything onto N and S we obtain (with F c = P c F and F s = P s F ) and Formally, we see later (cf. Lemma 22) that ψ is well approximated by the fast Ornstein-Uhlenbeck process Z s . Thus, we can eliminate ψ in (31) by explicitly averaging over the fast modes. Now the first main result of this paper is: Theorem 17 (Approximation I) Under Assumptions 2, 4, 6, 5, and 8, let u be a solution of (1) with splitting u = a + ψ defined in (30) with the initial condition u(0) = a(0) + ψ(0) with a(0) ∈ N and ψ(0) ∈ S where a(0) and ψ(0) are of order one, and b is a solution of (3) with b(0) = a(0). Then for all p > 0 and all κ ∈ (0, 1 2m+1 ), there exist a constant C > 0 such that where with fast OU-process Z s defined in (27) We see that the first part of (34) depending on the initial condition decays exponentially fast on the time-scale of order O(ε 2 ).

Corollary 18
If in the previous theorem we additionally assume that Assumption 3 holds and ψ(0) L mp n ≤ C for some C > 0, then we can replace T 1 ∧ τ * in (33) by T 1 .

Remark 19
In case of Corollary 18 we can bound the error even in L pm n .
4.2 Second case σ ε = 1 and α i,0 = 0 for i = 1, .., n In this case (25) takes the form Again (cf. (30)) we split the solution u into u(t, x) = a(t) + εψ(t, x). Plugging (30) into (35) and projecting everything onto N and S yields and where Z s (t) was defined in (27). We write (36) as Now, applying Taylor's expansion to the function F c i : L 2 (G) → R, yields the following stochastic limiting equation with error where The second main result of this paper is: Theorem 20 (Approximation II) Under Assumptions 2, 4, 5 and 9, let u be a solution of (1) with splitting u = a + εψ defined in (30) with the initial condition u(0) = a(0) + εψ(0) with a(0) ∈ N and ψ(0) ∈ S where a(0) and ψ(0) are of order one, and b is a solution of (5) with b(0) = a(0). Then for all p > 0, for sufficiently large ζ ≫ 1 and all κ ∈ (0, 1 m+2 ), there exists C > 0 such that In our examples if we assume E exp{cδ|b(0)| m−1 } ≤ C for some suitable c > 0 and for one δ > 0, then Assumption 9 is true. See Section 8.1.
Corollary 21 If in the previous theorem additionally Assumption 3 holds and ψ(0) L mp n ≤ C for C > 0, then we can replace T 1 ∧ τ * in (40) by T 1 .
The sufficiently large ζ depends mainly on κ and T 0 .

Bounds for the high modes
Let us summarize Equations (32) and (37) for ρ ∈ {0, 1} by In the first lemma of this section, we see that ψ is well approximated by the fast Ornstein-Uhlenbeck process Z s (cf. (27)).
Lemma 22 Under Assumption 5, there is a constant C > 0 such that for p ≥ 1 and κ > 0 from the definition of τ * E sup Proof. From (41) using semigroup estimates and Assumption 5 we obtain Lemma 23 Under Assumptions 2 and 4, for every κ 0 > 0 and p ≥ 1 there is a constant C, depending on p, α k , λ k , κ 0 and T 0 , such that where Z s (t) was defined in (27).
Using Identity (44) with Equation (45), we obtain: From the stochastic Fubini theorem, we obtain where Taking · p L p n on both sides of (46) and using (8), we obtain Using Hölder inequality with 1 p + 1 q = 1 for sufficiently large p implies Taking supremum after expectation, yields Hence by Definition of y i (47) where we used the definition of covariance operator (14). Hence, using the bounds on g j where we used (13). Thus Now, returning to Equation (48) and using Equation (50), yields We finish the proof by using Hölder inequality to derive for all p > 1 and sufficiently large q > 2 The following corollary states that ψ(t) is with high probability much smaller than ε −κ as assumed the Definition 10 for t ≤ τ * . We show later τ * ≥ T 0 with high probability.
Proof. By triangle inequality and Lemma 23, we obtain from (42) which implies (51) for κ < 2−ρ m . Let us now state a result similar to averaging. When we integrate over the fast decaying contribution of the initial condition in ψ, then this leads to terms of order O(ε 2 ).
6 Averaging over the fast OU-process Lemma 26 Let Assumption 4 hold and consider Z i,j (t) as defined in (29). Then for arbitrary δ 0 ∈ (0, 1 2 ) we obtain and Moreover, the O-terms are uniform in i, j, k and ℓ.
Proof. For the first part, we follow the same steps as in Lemma 23 to obtain Using Hölder inequality, we derive for sufficiently large q and for a constant independent on i and j E sup We finish the proof by fixing δ 0 = 2 pq < 1 2 for large q and p.
For the second part we use Cauchy-Schwarz inequality to obtain Using the first part, yields (53). In next corollary we state without proof the general case of Lemma 26. For the proof we can follow the same steps as in the proof of Lemma 26.
Corollary 27 Under the assumptions of Lemma 26 we have Lemma 28 Let the assumptions of Lemma 26 hold and let X be a real valued stochastic process such that for some small r ≥ 0 we have Again all O-terms are uniform in the indices ℓ j and k j .
Proof. For the first part, we use Itô isometry to obtain For the second part using Burkholder-Davis-Gundy theorem and Hölder inequality, yields By Gaussianity and the first part we obtain Analogously, for the last term Using Hölder, Gaussianity and the first part, we obtain (57).
In the following we state and prove the averaging lemma over the fast OUprocess Z i,j (cf. (29)).
Lemma 29 Under Assumption 2, 4 and 6, let X be as in Lemma 28 and N ≤ m. Then for N odd and for N even The O-terms are again uniform in all indices.
Let us state explicitly some A's appearing in the proof of the theorem.
Example 31 For N = 1 we have A ℓ k = 1 λ k q ℓ,ℓ k,k 1 2 , and for N = 2 For larger N the terms have similar structure, but there are about N/2 many.
Proof. Fix a small δ 0 < 1 N for N > 1. First, recall |X| = O(ε −r ). For the first part we treat N = 1 and 3. The general case follows by induction.
For N = 1 we apply Itô formula to the term XZ i,j to obtain
We prove the second part only for N = 2 and we can proceed by induction. Applying Itô formula to X · 2 i=1 Z ℓi,ki and integrating from 0 to t, we obtain Using Corollary 27 and Lemma 28 to obtain Xds.

Xds.
For N ∈ {4, 6, ..} we apply Itô formula to the term X Using Corollary 27 and Lemma 28 to obtain as in the odd case before The first factor in the sum is summable over j 1 and j 2 by Condition (13). Now, we can proceed by induction and apply the assertion for N − 2 to obtain (59).
Lemma 32 Under Assumption 2, 4 and 6 let X be as in Lemma 29. Then, for ℓ ∈ N n 0 with m ≥ |ℓ| ≥ 1, we obtain: 1-If one of the ℓ i is odd, then 2-If all ℓ i are even, then there is a constant C ℓ such that where C ℓ is given by Proof. From the definition of Z s (cf. (27), we obtain We focus in the proof on the case n = 1 and n = 2 as they are needed for our applications. The general case follows similarly but it is technically more involved. For n = 1 we have ℓ = ℓ 1 and Now we consider two cases. First if |ℓ| is odd, then Lemma 29 with N = |ℓ| yields And then as the A's are summable Secondly, if |ℓ| is even, then Lemma 29 implies As the A's are summable Xds +O(ε 1−r ).
For n = 2, we have N = |ℓ| = ℓ 1 + ℓ 2 and from Equation (66) with i 1 = · · · = i ℓ1 = 1 and i ℓ1+1 = · · · = i |ℓ| = 2. Similarly to the first part, we consider two cases. First if |ℓ| is odd, then we apply Lemma 29 to obtain As the A's are summable In the second case, when |ℓ| is even, we apply Lemma 29 and analogously to the first case we obtain We obtain We can distinguish between two cases when |ℓ| is even. First one of ℓ 1 and ℓ 2 is odd. Here q i ℓ 1 ,i ℓ 1 +1 j ℓ 1 ,j ℓ 1 +1 = 0, where i ℓ1 = 1 and i ℓ1+1 = 2. Thus In the second case when ℓ 1 and ℓ 2 are both even, we have The general case for n > 2 follows in a similar way, as the random variables where C ℓ was defined in (65) and the error is bounded byR = O(ε 1−2mκ−κ0 ).
Substituting from (68) into (36), yields Taylor's expansion for the polynomial F i : L p n → R yields where R (1) (t) is given by We see later that R (1) is small, as all terms contain at least one R = O(ε 2−mκ ). Taylor's expansion again for the polynomial F i : L p n → R, yields Applying the Averaging-Lemma 32, yields where C 0 = 1 and C ℓ = 0 if one ℓ i is odd. Thus To boundR we use Lemmas 25 and 23 and Assumption 5.
Definition 34 Define the set Ω * ⊂ Ω such that all the following estimates hold on Ω * sup and sup Proposition 35 Ω * has approximately probability 1. Proof.
Theorem 36 Assume that Assumptions 5 and 8 hold. Suppose a(0) = O(1) and ψ(0) = O(1). Let b be a solution of (3) and a as defined in (67). If the initial conditions satisfy a(0) = b(0), then for κ < 1 2m+1 we obtain and sup We note that all norms in a finite dimensional space are equivalent. Thus for simplicity of notation we always use the standard Euclidean norm.
Proof of Corollary 18. We note that by the semigroup estimate based on Assumptions 3 and Equation Thus from the Theorem 36 we derive Hence, Proceeding as in the proof of Theorem 17 we bound the error in L mp n .

Application of Approximation Theorem I
In this subsection we consider all examples with non-homogeneous Neumann boundary condition on [0, 1] 2 . Here the eigenfunctions are

Physical Application (Nonlinear Heat Eq.)
The heat equation plays a significant role in several areas of science including mathematics, probability theory and financial mathematics. In probability theory for instance, the heat equation is used for studying Brownian motion via the Fokker-Planck equation.
To apply our main Theorem 17, we consider the following nonlinear heat Equation with stochastic Neumann boundary condition.
If we choose for any µ > 0 that α 2 i,k ≤ C|k| −2µ for i = 1, . . . , 4 and all k ∈ N, then C 2 is finite and furthermore, all summability conditions are satisfied.
Let us finally check the bound on b. Taking the product with b on both sides of (84), yields 1 2 ∂ t |b| 2 = C|b| 2 − |b| 4 ≤ C|b| 2 .
Using Gronwall's lemma, we obtain for 0 ≤ t ≤ T 0 that Thus Assumption 8 is always true for deterministic initial conditions if we choose C 0 sufficiently large.

Chemical Application
A simple archetypical example for a reaction-diffusion system is a cubic autocatalytic reaction between two chemicals according to the rule A + B → 2B with rate r = ρu 1 u 2 2 . Denoting by u 1 and u 2 the concentration of A and B, respectively. The two species satisfy the equations: with respect to stochastic boundary conditions for i = 1, 2 where W ij (t) = ∞ k=1 α ij ,k β ij ,k (t)f k for j = 1, . . . , 4, and f k defined as before.
We note that high fluctuations in combination with fast diffusion lead to effective new terms describing the transformation of b 1 to b 2 . Although both terms individually do not change the average u i dx = b i , their nonlinear combination does.
Let us check the bound on b from Assumption 8. We note that Hence, for all times t > 0 we obtain b(t)

Proof of the Approximation Theorem II
In this section, we use many ideas and lemmas of the previous sections, as the main ideas are similar.
Lemma 37 Let Assumption 5 holds. Then for R (2) defined in (39) as Proof. Using Assumption 5 Definition 38 Define the set * * Ω ⊂ Ω such that for sufficiently large ζ ≫ 1 all the following estimates hold on * * and sup Proposition 39 * * Ω has approximately probability 1. Proof.

Application of Approximation Theorem II
In this subsection we apply our main Theorem 20 to the nonlinear heat equation (83) and a cubic auto-catalytic reaction (85) with σ ε = 1 and non-zero α k,0 .

Thus
So the probability is close, but not very close to 1, as δ cannot be arbitrarily large.