Stochastic Cahn-Hilliard Equation Research Articles

Analysis of a Mixed Finite Element Method for Stochastic Cahn-Hilliard Equation with Multiplicative Noise

Analysis of a Mixed Finite Element Method for Stochastic Cahn-Hilliard Equation with Multiplicative Noise

Averaging principles for multiscale stochastic Cahn–Hilliard system

In this paper, we will establish averaging principles for the multiscale stochastic Cahn–Hilliard system. The stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. Under suitable conditions, two kinds of averaging principle (the autonomous case and the nonautonomous case) are proved, and as a consequence, the multiscale system can be reduced to a single stochastic Cahn–Hilliard equation (averaged equation) with a modified coefficient, the slow component of multiscale stochastic Cahn–Hilliard system towards to the solution of the averaged equation in moment (the autonomous case) and in probability (the nonautonomous case).

Improved estimates for the sharp interface limit of the stochastic Cahn–Hilliard equation with space-time white noise

Improved estimates for the sharp interface limit of the stochastic Cahn–Hilliard equation with space-time white noise

Density convergence of a fully discrete finite difference method for stochastic Cahn–Hilliard equation

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn–Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in L 1 ( R ) L^1(\mathbb {R}) of the numerical solution. Our results partially answer positively to the open problem posed by J. Cui and J. Hong [J. Differential Equations 269 (2020), pp. 10143–10180] on computing the density of the exact solution numerically.

Moderate deviations for stochastic Cahn-Hilliard equations with a random dynamical boundary driven by Poisson random measures

This work concerns a stochastic Cahn-Hilliard equation with a random dynamical boundary condition driven by Poisson random measures. Due to the disturbance of pure jump Lévy noise both in the domain and on its boundary, we drive the tightness of deviation processes by employing the stochastic control argument and the splitting method. With the help of the weak convergence approach, we establish the moderate deviations of this system, which bridge the gap between the central limit approximation and large deviations.

Strong convergence rates of an explicit scheme for stochastic Cahn–Hilliard equation with additive noise

Strong convergence rates of an explicit scheme for stochastic Cahn–Hilliard equation with additive noise

Finite element approximation of the linearized stochastic Cahn–Hilliard equation with fractional Brownian motion

We perform a numerical analysis of the linearized stochastic Cahn–Hilliard equation driven by infinite-dimensional fractional Brownian motion with Hurst index H>12. The equation is discretized using a standard finite element method in space and a fully implicit backward Euler method in time. We prove strong convergence estimates for the considered stochastic Cahn–Hilliard equation. Finally, numerical experiments are performed to confirm the theoretical results.

Robust a posteriori estimates for the stochastic Cahn-Hilliard equation

We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation and a nonlinear random partial differential equation. The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as well as the intensity of the stochastic noise. We provide numerical simulations to demonstrate the practicability of the proposed adaptive algorithm.

Weak convergence of the backward Euler method for stochastic Cahn–Hilliard equation with additive noise

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen–Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn–Hilliard equation setting for the first time.

Higher moments for the stochastic Cahn–Hilliard equation with multiplicative Fourier noise

We consider in dimensions the ɛ-dependent stochastic Cahn–Hilliard equation with a multiplicative and sufficiently regular in space infinite dimensional Fourier noise with strength of order , γ > 0. The initial condition is non-layered and independent from ɛ. Under general assumptions on the noise diffusion σ, we prove moment estimates in H 1 (and in when d = 1). Higher H 2 regularity p-moment estimates are derived when σ is bounded, yielding as well space Hölder and bounds for , and path a.s. continuity in space. All appearing constants are expressed in terms of the small positive parameter ɛ. As in the deterministic case, in H 1, H 2, the bounds admit a negative polynomial order in ɛ. Finally, assuming layered initial data of initial energy uniformly bounded in ɛ, as proposed by Chen (1996 J. Differ. Geom. 44 262–311), we use our H 1 2d-moment estimate and prove the stochastic solution’s convergence to as a.s. when the noise diffusion has a linear growth.

A WELL-POSEDNESS RESULT FOR A STOCHASTIC CAHN-HILLIARD EQUATION

This paper is about the study of the well-posedness of a stochastic Cahn-Hilliard equation driven by white noise induced by a Q-Brownian motion. The proof of the existence of a unique global solution relies on the Galerkin method together with a monotonicity method.

Wellposedness and regularity estimates for stochastic Cahn–Hilliard equation with unbounded noise diffusion

Wellposedness and regularity estimates for stochastic Cahn–Hilliard equation with unbounded noise diffusion

An LDG Method for Stochastic Cahn-Hilliard Type Equation Driven by General Multiplicative Noise Involving Second-Order Derivative

An LDG Method for Stochastic Cahn-Hilliard Type Equation Driven by General Multiplicative Noise Involving Second-Order Derivative

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.

The stochastic Cahn–Hilliard equation with degenerate mobility and logarithmic potential

We prove existence of martingale solutions for the stochastic Cahn–Hilliard equation with degenerate mobility and multiplicative Wiener noise. The potential is allowed to be of logarithmic or double-obstacle type. By extending to the stochastic framework a regularization procedure introduced by Elliott and Garcke in the deterministic setting, we show that a compatibility condition between the degeneracy of the mobility and the blow-up of the potential allows to confine some approximate solutions in the physically relevant domain. By using a suitable Lipschitz-continuity property of the noise, uniform energy and magnitude estimates are proved. The passage to the limit is then carried out by stochastic compactness arguments in a variational framework. Applications to stochastic phase-field modelling are also discussed.

Conservative stochastic two-dimensional Cahn–Hilliard equation

We consider the stochastic two-dimensional Cahn–Hilliard equation which is driven by the derivative in space of a space-time white noise. We use two different approaches to study this equation. First we prove that there exists a unique solution Y to the shifted equation (1.4). Then X:=Y+Z is the unique solution to the stochastic Cahn–Hilliard equation, where Z is the corresponding O-U process. Moreover, we use the Dirichlet form approach in (Probab. Theory Related Fields 89 (1991) 347–386) to construct a probabilistically weak solution to the original equation (1.1) below. By clarifying the precise relation between the two solutions, we also get the restricted Markov uniqueness of the generator and the uniqueness of the martingale solutions to the equation (1.1). Furthermore, we also obtain exponential ergodicity of the solutions.

Well-posedness and large deviation principle for a class of SPDEs

In this paper, we study a class of stochastic partial differential equations with general locally monotone coefficients under thevariational framework. By proving the existence of martingale solutions and pathwise uniqueness, we first get the existence and uniqueness of strong solutions. Then we prove the large deviation principle by using the stochastic control and weakconvergence approach. The mainresults are applicable to various concrete SPDE models, such as stochastic Cahn-Hilliard equation, stochastic Kuramoto-Sivashinsky equation and stochastic 3D tamedNavier-Stokes equation.

Absolute continuity and numerical approximation of stochastic Cahn–Hilliard equation with unbounded noise diffusion

In this article, we consider the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by using the strong convergence of the spectral Gakerkin approximation. Then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient, which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1/3<α<1. To approximate the density function of the exact solution, we propose a full discretization based on the spatial spectral Galerkin approximation and the temporal drift implicit Euler scheme. Furthermore, a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method. Consequently, we obtain the sharp mean square convergence rates in both time and space via Sobolev interpolation inequalities and semigroup theories. To the best of our knowledge, this is the first result on the convergence rate of full discretizations for the considered equation.

Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises

We study the asymptotic properties of the stochastic Cahn-Hilliard equation with the logarithmic free energy by establishing different dimension-free Harnack inequalities according to various kinds of noises. The main characteristics of this equation are the singularities of the logarithmic free energy at 1 and −1 and the conservation of the mass of the solution in its spatial variable. Both the space-time colored noise and the space-time white noise are considered. For the highly degenerate space-time colored noise, the asymptotic log-Harnack inequality is established under the so-called essentially elliptic conditions. And the Harnack inequality with power is established for non-degenerate space-time white noise.

Large deviations for a stochastic Cahn–Hilliard equation in Hölder norm

We consider a stochastic Cahn–Hilliard equation driven by a space–time white noise. We prove that the law of the solution satisfies a large deviations principle in the Hölder norm. Our proof is based on the weak convergence approach for large deviations.