Sharp Interface Limit of Stochastic Cahn-Hilliard Equation with Singular Noise

We study the sharp interface limit of the two dimensional stochastic Cahn-Hilliard equation driven by two types of singular noise: a space-time white noise and a space-time singular divergence-type noise. We show that with appropriate scaling of the noise the solutions of the stochastic problems converge to the solutions of the determinisitic Mullins-Sekerka/Hele-Shaw problem.


Introduction
We consider the stochastic Cahn-Hilliard equation with additive noise where 0 1 2 , is the outward unit normal to , 0 is a constant, 0 is a small parameter and is a singular space-time noise which will be specified later on. The nonlinearity in Eq. 1.1 is taken as where (1. 3) The Cahn-Hilliard equation is a model for the non-equilibrium dynamics of metastable states in phase transitions [6,15,17]. The parameter in Eq. 1.2 represents an "interaction length", which is typically very small, and is an order parameter (scaled concentration) which assumes the values 1 and 1, respectively, in the regions occupied by the pure phases. The phase separation consists of two stages a so-called spinodal decomposition which is followed by a coarsening process. Starting from a fully mixed state, e.g., a random perturbation around the initial mass, the system undergoes a short phase, so-called spinodal decomposition, during which the initial phases are formed. The solution quickly approaches the respective values 1, 1 in the regions occupied by the pure phases. The pure phases are separated by a thin region with a width proportional to , so-called diffuse interface. Once the diffuse interface is fully formed, the evolution enters a second stage, so-called coarsening phase, during which the originally fine-grained structure coarsens, the geometric structure of the phase regions gradually becomes simpler and eventually tends to regions of minimum surface area with preserved volume.
A rigorous sharp interface limit of the deterministic Cahn-Hilliard (1.3) has been obtained in [1] Under the assumption that the interfaces have been formed, i.e., that there exists a smooth closed curve 00 such that 0 1 in , in the region enclosed by 00  d , is the mean curvature of with the sign convention that convex hypersurfaces have positive mean curvature, is the normal velocity of the interface with the sign convention that the normal velocity of expanding hypersurfaces is positive, 0 1 is the normal to and , are respectively the restriction of on (the exterior/interior of in ).
The sharp interface limit of stochastic Cahn-Hilliard equation with trace-class noise has been studied in [2]. There, the authors show that for sufficiently large the sharp interface limit of Eq. 1.2 satisfies the deterministic Mullins-Sekerka/Hele-Shaw problem (1.4). In the recent paper [3] the authors show convergence of structure preserving numerical approximation of the stochastic Cahn-Hilliard to the deterministic Mullins-Sekerka/Hele-Shaw problem. In addition [3] obtains a uniform convergence result which implies convergence of the zero-level set of the numerical solution to the free boundary 0 of Eq. 1.4. The case of 1 remains an open problem. In this paper we study the sharp interface limit of stochastic Cahn-Hilliard equation driven by singular noise. We consider the Cahn-Hilliard-Cook model, proposed by Cook, cf. [6] and [15]), which incorporates thermal fluctuations in the form of an additive noise in Eq. 1.2. We choose the noise as 1 or 2 , where 1 is mass-conserved 2 -cylindrical Wiener process and 2 is an 2 2 -cylindrical Wiener process. We note that in the latter case the equation is known as the time-dependent Ginzburg-Landau (TDGL) equation and is also related to the stochastic quantization for 4 2 -quantum field. For the existence and uniqueness results for these two kinds of equations we refer to [8,22] and the reference therein.
To analyze the sharp interface limit of the solution to Eq. 1.2 for the case of the space-time white noise 1 , we adapt the approach of [2]. We estimate the difference of to an approximate solution which is constructed by the matched asymptotic expansion method such that the interface is the zero level set of , cf. [1]. The approximation satisfies a perturbed equation Since the goes to zero as 0, we can show that for 0 the differences , converge to 0 for estimate . Hence the arguments of [2] are not directly transferable to our case. Instead, we make use of the idea of Da Prato-Debussche [10]: after introducing a variable 0 2 we study the translated difference which enjoys better regularity properties. By combining the estimates for and we bound the error and obtain the sharp-interface limit. For the case singular divergence-type noise 2 the Eq. 1.2 is illposed in the classical sense, since the solution is not a function but a distribution. Hence, it does not make sense to consider the sharp interface limit of Eq. 1.2 directly. Instead we follow the renormalization approach: we employ a suitable regularization 2 of the the space-time white noise 2 and consider the regularized equation: where 3 is a renormalization term (see Eq. 5.7) which ensures that converges to for 0, where is the unique solution of the renormalized version of Eq. 1.2, see Eq. 5.11. The analysis in the case of the divergence-type noise is complicated by the fact that for fixed 0 the renormalization constant in Eq. 1.6 diverges, i.e., that as 0. By choosing for some 0 and goes to 0 (see Theorem 5.6) the constant becomes small as 0 which enables us to control the term . The remaining steps in the analysis of Eq. 1.6 are analogical to the first case: we obtain that the sharp interface limit of Eq. 1.6 for 26 3 is the deterministic Mullins-Sekerka problem (1.4). The paper is organized as follows. In Section 2 we give an overview of existing results on sharp interface limits for related problems. In Section 3, we introduce the notation and state preliminary results. The sharp interface limit for the space-time white noise is stated in Section 4 and we prove it in Section 4.1. In Section 5 we use a similar argument as we used in Section 4.1 to prove the results for divergence-type noise.

Overview of Existing Results
For stochastic Cahn-Hilliard eqaution, the authors in [2] prove that for large the sharp interface limit of Eq. 1.2 also satifies the deterministic Hele-Shaw model if is a traceclass noise. For 1, the sharp interface limit is also conjectured to satisfy the following stochastic Hele-Shaw model: In [4], the authors prove that the sharp interface limit of generalized Cahn-Hilliard equation: (2. 3) It is well-known that the movement of interface is characterized by mean curvature flow (see e.g. [9,11,16]). Unlike the solution to the Allen-Cahn equation, the solution to the Cahn-Hilliard (1.3) does not approach 1 away from the interface exponentially fast. The direct application of the method of asymptotic matching in [9] does not lead to the desired approximation solutions. In stochastic case which is also called Model A of [15]), the authors in [12] and [26] consider the following stochastic Allen-Cahn equation The noise is constant in space and smooth in time. For 0 the correlation length goes to zero at a precise rate and 0 converges to a Brownian motion pathwisely. They prove that the dynamics of the phase-separating hyperplane appearing in the limit is given by stochastic mean curvature flow (see also in [13,Chapter 4]). For space-time white noise, in [25] the authors prove the "exponential loss of memory property". But for sharp interface limit, there is still no result for space-time white noise.

Notations and Preliminaries
Throughout the paper, we use the notation if there exists a constant 0 which is independent to and time such that . If is depend on , we use the notation . We write if and . In the rest of this paper, we use the notation to represent for simiplicity. Moreover for any , we can define a bounded operator   in . (4.1) We assume that the interface has been formed initially. That is, there exists a smooth closed curve 00 such that 0 1 in , the region enclosed by 00 , and 0 1 in 00 .
Our main theorem will show that as 0, tends to , which, together with a free boundary 0 , satisfies the deterministic Hele-Shaw problem (1.4). We present now the following spectral estimate which is useful in our proof.   . ..

Since
, we obtain the assertion.

The Decomposition of the Equation for the Error
On Combining (4.1), (1.5) and noting (4.2) we obtain 2 1 0 on . for some 1. Here we require that 1 2 2 0 2 2 0 that is 1 2 4 (4.15) which can be obtained by choosing small enough 0. Hence by Eqs. 4.7 and 4.8, we obtain that for any 1 2 4 This implies that for any 2 1 and small enough 0, 4 2 . (4.16) Hence the statement follows by Cheybeshev's inequatliy.

Local-in-time estimate for Y up to T on the set ,
In the remainder of the proof we fix where is defined in Lemma 4.6 and work pathwise. We note that by definition 2 for . By taking inner product with 1 in both side of Eq. 4.5 we have that where we used Hölder's inequality in the first inequality and Lemma 4.6 in the last inequality.
It has been proved in [1, Lemma 2.2] that 3 for in a bounded set which is the case in the lines below. Then Then for any we have   i.e. A direct calculation yields that Similarly as above 2 6 2 .
Since are uniformly bounded in and 0 1 , we have that  The statement of the Theorem 4.3 then follows on combining the above inequality with Eq. 4.28.

Sharp Interface Limit for the Divergence-Type Noise
Throughout this section we consider the singular divergence-type noise , where is an 2 , i.e. 1 2 . There exist two independent 2 0 -cylindrical Wiener processes 1 and 2 such that 1 2 . Similarly as in [22,23], it follows that the solution to Eq. 1.2 with the divergence-type noise is distribution-valued. It does not appear to be possible to obtain the sharp-interface limit by directly considering (5.11). Thus we study the sharp interface limit of the regularized (1.6) instead.

Existence and Uniqueness of Solutions to Eq. 1.6
In order to consider the convolution of the noise with an approximate delta function (the standard mollifier). we need to extend the noise to the whole space 2 . Considering the Neumann boundary condition, it is reasonable to extend it evenly to 1 1 2 first, then do a periodical extension to the whole space. That is, for any function on which satisfies the Neumann boundary condition, we view it as a function on 2 by For simplicity we write

Since
, similar as in the proof in [19,22,23], for any 1 2 3, as 0, converges in 0 for any 0 whose limit is denoted as . Here is defined as the Besov space , see [23] and the reference therein for details. Then we denote Remark 5. 3 We note that in [22] the authors consider the periodical boundary condition, which is different from the Neumann boundary condition. But by our extension method as we explained before, a similar proof follows.
In fact, lim 0 in 0 1 0 . Let , also converges to in 0 1 , which is the unique solution to where is defined in Eq. 5.10.

The Sharp Interface Limit of Eq. 1.6
Similarly as in the proof of Theorem 4.3 we prove that for a suitable choice , the solutions to Eq. 5.11 will converge to the solution to deterministic Hele-Shaw model (1.4).
The method is a modification of the one in Section 4.  Proof We proceed similarly as in Section 4. where is the interior of in .
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