A stochastic mass conserved reaction-diffusion equation

In this paper, we prove a well posedness result for an initial boundary value problem for a stochastic nonlocal reaction-diﬀusion equation with nonlinear diﬀusion together with a nul-ﬂux boundary condition in an open bounded domain of R n with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion


Introduction
We study the problem (P ) x ∈ D, t ≥ 0 A(∇ϕ).ν= 0, on ∂D × R + ϕ(x, 0) = ϕ 0 (x), x ∈ D where: • D is an open bounded set of R n with a smooth boundary ∂D; • ν is the outer normal vector to ∂D; • The initial function ϕ 0 is such that ϕ 0 ∈ L 2 (D); • We suppose that the nonlinear function f is a smooth function which satisfies the following properties: • We assume that A = ∇ v Ψ(v) : R n → R n for some strictly convex function Ψ ∈ C 1,1 (i.e.Ψ(v) ∈ C 1 (R n ) and ∇Ψ(v) is Lipschitz-continuous) satisfying for some constant c 1 > 0. We remark that (1.1) implies that for all a, b ∈ R n , where C is a positive constant, and that the strict convexity of Ψ implies that A is strictly monotone, namely there exists a positive constant C 0 such that (A(a for all a, b ∈ R n . We remark that if A is the identity matrix, the nonlinear diffusion operator − div(A(∇u)) reduces to the linear operator −∆u.
• The function W = W (x, t) is a Q-Brownian motion.More precisely, let Q be a nonnegative definite symmetric operator on L 2 (D), {e l } l≥1 be an orthonormal basis in L 2 (D) diagonalizing Q, and {λ l } l≥1 be the corresponding eigenvalues, so that Qe l = λ l e l for all l ≥ 1.Since Q is of trace-class, it follows that for some positive constant Λ 0 .We suppose furthermore that e l ∈ H 1 (D) ∩ L ∞ (D) for l = 1, 2... and that there exist positive constants Λ 1 and Λ 2 such that and (1.6) Let (Ω, F, P) be a probability space equipped with a filtration (F t ) and {β l (t)} l≥1 be a sequence of independent (F t )-Brownian motions defined on (Ω, F, P); the Q-Wiener process W is defined by in L 2 (D).We recall that a Brownian motion β(t) is called an (F t ) Brownian motion if it is (F t )-adapted and the increment β(t) − β(s) is independent of F s for every 0 ≤ s < t.
We define : where • is the norm corresponding to the space H.We also define •, • Z * ,Z as the duality product between Z and its dual space The corresponding deterministic equation in the case of linear diffusion, when A is the identity matrix, has been introduced by Rubinstein and Sternberg [17] as a model for phase separation in a binary mixture.The well-posedness and the stabilization of the solution for large times for the corresponding Neumann problem were proved by Boussaïd, Hilhorst and Nguyen [4].They assumed that the initial function was bounded in L ∞ (D) and proved the existence of the solution in an invariant set using a Galerkin approximation together with a compactness method.The interfacial evolution process corresponding to a second order mass conserved Allen-Cahn equation shares many properties with the fourth order Cahn-Hilliard equation as discussed in [17].Da Prato and Debussche proved the existence and the uniqueness of the solution of a stochastic Cahn-Hilliard equation in [6] with an additive space-time white noise.
In our work, inspired by this paper, we introduce a nonlinear stochastic heat equation, perform a change of functions in order to maintain a "deterministic style" mass conserved equation by hiding the noise term and prove the existence of the solution in suitable Sobolev spaces similar to those in [6].Funaki and Yokoyama [8] derive a sharp interface limit for a stochastically pertubed mass conserved Allen-Cahn equation with a sufficiently mild additive noise.This is different from the stochastic term in this paper which is not smooth.A singular limit of a rescaled version of Problem (P) with linear diffusion has been studied by Antonopoulou, Bates, Blömker and Karali [1] to model the motion of a droplet.However, they left open the problem of proving the existence and uniqueness of the solution, which we address here.The problem that we study is more general then the one in [1] since it has a nonlinear diffusion term.The proof is based on a Galerkin method together with a monotonicity argument similar to that used in [14] for a deterministic reaction-diffusion equation, and that in [12] for a stochastic problem.
Our paper is organised as follows.In section 2 an auxiliary problem is introduced, more precisely the nonlinear stochastic heat equation and a change of function is defined to obtain an equation without the noise term; this simplifies the use of the Galerkin method in section 3, which yields uniform bounds for the approximate solution in L ∞ (0, T ; L 2 (Ω × D)), L 2 (Ω × (0, T ); H 1 (D)) and in L 2p (Ω × (0, T ) × D).We deduce that the approximate weak solution weakly converges along a subsequence to a limits.The main problem is then to identify the limit of the elliptic term and the reaction term, which we do by means of the so-called monotonicity method.We prove in section 4 the uniqueness of the weak solution which in turn implies the convergence of the whole sequence.Finally, in section 5 we return to the study of the nonlinear stochastic heat equation and prove the existence and uniqueness of the solution.

A preliminary change of functions
We consider the Neumann boundary value problem for the stochastic nonlinear heat equation Krylov and Rozovskii [12] proved the well-posedness result for a classes of problems similar to Problem (P 1 ) using a definition of solution in the distribution sense, while Gess [10] defines a solution in the sense of L 2 (D), namely almost everywhere in D.More precisely, he defines a strong solution as follows (cf.[10], Definition 1.3).

Definition 2.1. (Strong solution)
We say that W A is a strong solution of Problem (P 1 ) if : (iv) W A satisfies a.s.for all t ∈ (0, T ) the problem A(∇W A (t)).n = 0, in a suitable sense of trace on ∂D. (2.1) We will show in Section 5 the existence and uniqueness of the strong solution W A of Problem (P 1 ).Moreover we will prove that We perform the change of functions then ϕ is a solution of (P) if and only if u satisfies: We remark that (P 2 ) has the form of a deterministic problem; however it is stochastic since the random function W A appears in the parabolic equation for u.
Definition 2.2.We say that u is a solution of Problem (P 2 ) if : (ii) u satisfies almost surely the problem: for all t ∈ [0, T ] In order to check the conservation of mass property, namely that and take the duality product of (2.2) with 1 for a.e.t and ω.
3 Existence of a solution of Problem (P 2 ) The main result is the following Theorem 3.1.There exists a unique solution of Problem (P 2 ).
Proof.In this subsection we apply the Galerkin method to prove the existence of a solution of Problem (P 2 ).Denote by 0 < γ 1 < γ 2 ≤ ... ≤ γ k ≤ ... the eigenvalues of the operator −∆ with homogeneous Neumann boundary conditions, and by w k, k = 0, ... the corresponding unit eigenfunctions in L 2 (D).Note that they are smooth functions.Proof.We check below that D w j (x)dx = 0 for all j = 0. Indeed, which implies that D w j w 0 dx = 0 for all j = 0.Moreover, it is standard that the eigenfunctions corresponding to different eigenvalues are orthogonal.
We look for an approximate solution of the form for all w j , j = 1, ..., m.We remark that u m (x, 0 Problem (3.2) is an initial value problem for a system of m ordinary differential equations with the unknown functions u im (t), i = 1, .., m so that it has a unique solution u m on some interval (0, T m ), T m > 0; in fact the following a priori estimates show that this solution is global in time.
First we remark that the contribution of the nonlocal term vanishes.Indeed for all j = 1, ..., m Therefore (3.2) reduces to the equation: We multiply (3.3) by u jm = u jm (t) and sum on j = 1, ..., m: Next we apply the monotonicity property of A (1.3) to bound the generalized Laplacian term, which yields 1 2 Using the property (F 1 ) we deduce that which we substitute in (3.5) to obtain :

A priori estimates
In what follows, we derive a priori estimates for the function u m .
Lemma 3.2.There exists a positive constant C such that Proof.Integrating (3.6) from 0 to t and taking the expectation we deduce that for all t ∈ [0, T ] where we have used (2.2).We deduce that : Therefore Using the property (F 2 ) we deduce that by (3.9) and (2.2), where c 3 is a positive constant.
Finally we show that the elliptic term is bounded in (H 1 (D)) .We have that Next we use (1.2) and (1.1) to estimate the term on the right-hand-side of (3.12) The last line follows from the a priori estimates and the regularity of the solution of Problem (P 1 ).
Hence there exist a subsequence which we denote again by {u m − M } and a function as m → ∞.
Next, we pass to the limit as m → ∞.
To that purpose we integrate in time the equation (3.3) to obtain Let y = y(ω) be an arbitrary bounded random variable, and let ψ be an arbitrary bounded function on (0,T).We multiply the equation (3.17) by the product yψ, integrate between 0 and T and take the expectation to deduce for all j=1,..,m.Next we pass to the limit in (3.18); we only give the proof of convergence for the last term using the a priori estimates and Hölder inequality.We have that This shows that |ψ(t)E t 0 D f (u m + W A )yw j dxds| is uniformly bounded by a function belonging to L 1 (0, T ).In addition using (3.15) we have that χyw j dxds for a.e.t ∈ (0, T ).Applying Lebesgue-dominated convergence theorem we deduce that : Performing a similar proof for each term in (3.18), we pass to the limit by using Lebesgue-dominated convergence theorem.This yields χw j dxds}dt, for all j = 1, .., m.
We remark that the linear combinations of w j are dense in V ∩ L 2p (D), so that s., it follows by applying Lemma 1.2 p.260 in [18] that u − M ∈ C(0, T ; H) a.s.
It remains to prove that : We do so by means of the monotonicity method.

Monotonicity argument
Let w be such that w Let c be a positive constant which will be fixed later.We define and prove below the following result Lemma 3.4.
Proof.First we estimate J 1 and apply (1.3) (F 3 ) and the mean value theorem yield: Choosing c ≥ 2C 4 , we conclude the result.
We write O m in the form where We integrate the equation (3.3) between 0 and T to obtain Next we recall a chain rule formula, which can be viewed as a simplified Itô's formula.
Proposition 3.1.Let X be a real valued function such that and suppose that h is measurable in time such that h ∈ L 1 (0, T ).Suppose that the function Then for all t ∈ [0, T ] In what follows, we will use the identity Taking the expectation of the equation (3.26) yields From this we obtain where δ = lim On the other hand, the equation (3.20) implies that Next we recall a second variant of the chain rule formula, which can be viewed as a simplified Itô's formula as in [15] [p.75 Theorem 4.2.5], and involves different function spaces.Consider the Gelfand triple where Z = V ∩ L 2p (D) and Z * are defined in the introduction.
Suppose that the function which we combine with (3.28) to deduce that (3.31)It remains to compute the limit of O 2 m : In view of (3.13), (3.15) and (3.16), we deduce that Combining (3.31) and (3.32), and remembering that O m ≤ 0, yields We obtain the inequality : Dividing by λ and letting λ → 0, we find that : Since v is arbitrary, it follows that a.s.a.e. in D × (0, T ).Taking the duality product of (3.33) with w ∈ V ∩ L 2p (D) we obtain that This completes the identification of the limit terms by the monotonicity method.
Next, we prove that u satisfies the equation (2.3) in Definition 2.2.We define The equation (3.35) implies that a.s. in for all t ∈ [0, T ].
In order to identify the last term of (3.36), we take its duality product ., .V * ,V with 1.
Remembering that the equation is mass conserved, we obtain 4 Uniqueness of the solution of Problem (P 2 ) Let ω be given such that two pathwise solutions of Problem (P 2 ), u 1 = u 1 (ω, x, t) and u 2 = u 2 (ω, x, t) satisfy in L 2 ((0, T ); V * ) + L 2p 2p−1 ((0, T ) × D).We take the duality product of this equation with where we remark that since In view of (1.3), (4.1) becomes for all t ∈ (0, T ).In addition, the property (F 3 ) implies that which in turn implies by Gronwall's Lemma that 5 Existence and uniqueness of the solution of Problem (P 1 ) In this section we return to the study of the solution W A of Problem (P 1 ), and derive a priori estimates for a Galerkin approximation in L ∞ (0, T ; L 2 (Ω×D))∩L 2 (Ω×(0, T ); H 1 (D))∩ L 2 (Ω × (0, T ); H 2 (D)) following an idea due to Gess [10].We are then in a position to show that W A is also bounded in L ∞ (0, T ; L q (Ω × D)) for all q ≥ 2, which is necessary for the proof of Lemma 3.2.
We show below a priori estimates, which imply that the elliptic term div(A(∇W A )) is bounded in L 2 (D) having in mind that Problem (P 1 ) is a special case of Problem (4.33) in [10] ( see also equation (2.8) in [10]).Whereas Gess concentrates on the special case of the p-Laplacian, we are interested in the uniformly parabolic case, which corresponds to m = 2 in [10] p.280-281.We also remark that there are no reaction terms i.e. f i = 0 for all i from 1 to n and that the noise is additive.However, Gess assumes that the nonlinear function Ψ is twice continuously differentiable while we only suppose that Ψ ∈ C 1,1 (R n ).
We prove the following result.
Proof.To begin with, we approximate the function Ψ by a sufficiently smooth function and ∇Ψ n (0) = 0, and derive a priori estimates for a Galerkin approximation as in [10] Note that (cf.[5] p.193) and that (cf.[10] Remark 2.3) This implies in particular that P m a → a, in L 2 (D) as m → ∞. (5.5) In addition, we have that (cf.[12] p. 49) Lemma 5.1.There exists a positive constant K such that (5.10) Proof.We first recall Itô's formula as in [16] p.16-17 which is based on [11] [p.153, Theorem 3.6], and is applicable to systems of stochastic ordinary differential equations .
Lemma 5.2.For a smooth vector function h and an adapted process (g(t), t ≥ 0) with where h is a vector of components h l , l = 1, .., m and dW is a vector of components dβ l , l = 1, .., m with β l a one-dimensional Brownian motion.Then, for F twice continuously differentiable in X and continuously differentiable in t, one has P m λ l e l dβ l (s) and h l = P m √ λ l e l , supposing that F does not depend on time and setting A (s))).We remark that in this case F does not depend on t.After integrating on D, we obtain almost surely, for all t ∈ [0, T ], Substituting (5.6) into (5.12)we obtain, (5.13) Taking the expectation, we obtain where we have used the fact that 2E[ We deduce from (5.3) that Taking the supremum of equation (5.14) and substituting (5.15) into (5.14)we obtain sup t∈(0,T ) This completes the proof of (5.10).
In order to obtain an H 2 -type estimate for W m A , we take the gradient of the equation (5.2).For all x ∈ D, we have that (5.16) We fix x ∈ D and apply below for a second time Itô's formula Lemma 5.2 to the integral equation (5.16)where in this case hdW = m l=1 ∇{P m λ l e l }dβ l (s) and with: A (x, t)), and g(s) = ∇{P m div(∇Ψ n (∇W m,n A (x, s)))} After integrating over D, we obtain almost surely, for all t ∈ [0, T ], In view of (1.1) and (5.6) we have that (5.17) Thus taking the expectation of (5.17) and using the fact that Adding (5.14) and (5.18), using (5.3), (1.5) and (1.6) we obtain where ).In view of (1.3) we obtain, which completes the proof of (5.7), (5.8) and (5.9).
Hence there exist a subsequence which we denote again by W m,n A and a function In addition, one can show the following result.
λ l e l β l (t), in L ∞ ((0, T ); L 2 (Ω; L 2 (D))). (5.22) Proof.For all t ∈ [0, T ], By [9] p. 20 we deduce that W 1 → 0 in C([0, T ]) as m → ∞.For W 2 , by the properties of the Brownian motion, we have that In order to prove that the right-hand side of (5.23) tends to zero as m → ∞, we use (5.3) and (5.5) to deduce that Let ε > 0 be arbitrary.We choose K such that P 2 ≤ ε 2 .For a fixed K, we choose m sufficiently large such that Let y be an arbitrary bounded random variable, and let ψ be an arbitrary bounded function on (0,T).Next we multiply the equation (5.2) by the product yψ, integrate on D between 0 and T and take the expectation to obtain P m ( λ l e l )β l (t)w j dx}dt.

Monotonicity argument
Let w be such that w ∈ L 2 (Ω × (0, T ); H 1 (D)) and let c be a positive constant.We define We will check as before the following result Lemma 5.4.O mn ≤ 0.
Proof.Using (5.6) and (1.3) we have that (5.28) We apply Ito formula Lemma 5.2 on (5.2) with F (X, t) = e −ct (X) 2 and F t = −ce −ct (X) 2 .After integrating on D and taking the expectation, we obtain almost surely, for all t ∈ [0, T ], Letting m and n tend to infinity in (5.30), we deduce that lim m,n→∞ where On the other hand, the equation (5.26) implies that a.s. in L 2 (D) Next we recall a simplified form of the Itô's formula given by [7] ( Theorem 4.32 p.106), which will suffice for our purpose.We do so since the Itô's formula given in Lemma 5.2 only applies to finite dimensional problems.
Lemma 5.5.Let h be an L 2 (D)-valued progressively measurable Bochner integrable process.Consider the following well defined process : Assume that a function F : [0, T ] × L 2 (D) → R and its partial derivatives F t , F x , F xx are uniformly continuous on bounded subsets of [0, T ] × L 2 (D), and that F (X(0), 0) = 0.Then, a.s., for all t ∈ [0, T ], where we note that T rA Ae l , e l L 2 (D) is bounded linear operator on L 2 (D).
Next, we prove below the boundedness of W A in L ∞ (0, T ; L q (Ω × D)), for all q ≥ 2. The proof of this result is based on an article by Bauzet, Vallet, Wittbold [2] where a similar result was proved for a convection-diffusion equation with a multiplicative noise on R n involving a standard adapted one-dimensional Brownian motion.More precisely, we follow the proof of Proposition A.5 of [2].Theorem 5.2.Let W A be a solution of Problem (P 1 ); then W A ∈ L ∞ (0, T ; L q (Ω × D)), for all q ≥ 2.
Proof.For each positive constant k, denote by Φ k : R → R the function 2 ξ 2 for all ξ ∈ R + .This yields in view of Definition 2.1 (i) that, Lemma 5.6.(i) One has 0 ≤ Φ k (ξ) ≤ c k for all ξ ∈ R where c k is a positive constant depending on k.
Next we apply Lemma 5.5 to (2.1), supposing that F does not depend on time and setting (5.36) Taking the expectation of (5.36), and using the fact that Φ k ≥ 0, we deduce from the fact from the coercivity property (1.3) and from (1.5) that Then using Lemma 5.6 (ii) and Gronwall Lemma we obtain, defining C(q) = 1 2 q(q − 1), ≤ C(q)Λ 1 t|D|e C(q)Λ 1 t .
Thus, E D Φ k (W A )dx is bounded independently of k.
Finally, since Φ k (W A (x, t)) converges to |W A (x, t)| q for a.e.x and t when k goes to infinity, if follows from Fatou's Lemma that ≤ C(q)Λ 1 t|D|e C(q)Λ 1 t for all t > 0. Therefore, W A ∈ L ∞ (0, T ; L q (Ω × D)) for all q ≥ 2.
Proof.By induction, we first prove that (A.4) is true for q = 1.
Using Hölder inequality, we deduce that We suppose that (A.4) is true for q = 2p − 3 and prove that it remains true for q = 2p − 2: Using Hölder inequality, we obtain

Lemma 3 . 1 .
The functions {w j } are an orthonormal basis of L 2 (D) and satisfy : D w j w 0 dx = 0 for all j = 0 and w 0 = 1 |D| .

O 2 mn where O 1 mn = E T 0 eA 2 }
.27) which completes the proof.We write O mn in the formO mn = O 1 mn + −cs {2 P m [div(∇Ψ n (∇W m,n A ))], W m,n A − c W m,n ]ds.