Dirk Blömker

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

Abstract: This paper gives theoretical results on spinodal decomposition for the stochastic Cahn–Hilliard–Cook equation, which is a Cahn–Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size.In more detail, our results can be summarized as follows. The Cahn–Hilliard–Cook equation depends on a small positive parameter ε which models atomic scale interaction length. We quantify the behavior of solutions as ε→ 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace ?ε is high as long as the solution stays within distance rε=O(εR) of the homogeneous state. The subspace ?ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent R depends on both the strength and the regularity of the noise.

Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation

The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arising in weather forecasting. To gain understanding of filters in applications such as these, it is hence of interest to study their behaviour when applied to infinite dimensional dynamical systems. This motivates the study of the problem of accuracy and stability of 3DVAR filters for the Navier–Stokes equation. We work in the limit of high frequency observations and derive continuous time filters. This leads to a stochastic partial differential equation (SPDE) for state estimation, in the form of a damped-driven Navier–Stokes equation, with mean-reversion to the signal, and spatially-correlated time-white noise. Both forward and pullback accuracy and stability results are proved for this SPDE, showing in particular that when enough low Fourier modes are observed, and when the model uncertainty is larger than the data uncertainty in these modes (variance inflation), then the filter can lock on to a small neighbourhood of the true signal, recovering from order one initial error, if the error in the observed modes is small. Numerical examples are given to illustrate the theory.

Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation

We consider the dynamics of a nonlinear partial differential equation perturbed by additive noise. Assuming that the underlying deterministic equation has an unstable equilibrium, we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium. More precisely, we show that most trajectories starting at the unstable equilibrium are driven away in two stages. After passing through a cylindrical region, most trajectories diverge from the deterministic equilibrium through a cone-shaped region which is centered around a finite-dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation, and on which the influence of the nonlinearity is surprisingly small. This abstract result is then applied to explain spinodal decomposition in the stochastic Cahn-Hilliard-Cook equation on a domain G. This equation depends on a small interaction parameter e > 0, and one is generally interested in asymptotic results as e → 0. Specifically, we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach e -2+dim G/2 with respect to the H 2 (G)-norm.

Amplitude equations for stochastic partial differential equations

: Formal Derivation of Amplitude Equations Rigorous Results on Bounded Domains, SDEs as Amplitude Equations Applications: Approximation of Invariant Measures Pattern Formation Below Criticality Approximate Center Manifold Large Domains, SPDEs as Amplitude Equations.

MARKOVIANITY AND ERGODICITY FOR A SURFACE GROWTH PDE

The paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach. We prove existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under nondegeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure.

Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities

In this paper we derive rigorously amplitude equations for stochastic partial differential equations with quadratic nonlinearities, under the assumption that the noise acts only on the stable modes and for an appropriate scaling between the distance from bifurcation and the strength of the noise. We show that, due to the presence of two distinct timescales in our system, the noise (which acts only on the fast modes) gets transmitted to the slow modes and, as a result, the amplitude equation contains both additive and multiplicative noise. As an application we study the case of the one-dimensional Burgers equation forced by additive noise in the orthogonal subspace to its dominant modes. The theory developed in the present paper thus allows us to explain theoretically some recent numerical observations on stabilization with additive noise.

Modulation Equations: Stochastic Bifurcation in Large Domains

We consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. We show that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a stochastic Ginzburg-Landau equation. We then proceed to show that this approximation also extends to the invariant measures of these equations.

Amplitude equations for SPDEs with cubic nonlinearities

For a quite general class of stochastic partial differential equations with cubic nonlinearities, we derive rigorously amplitude equations describing the essential dynamics using the natural separation of timescales near a change of stability. Typical examples are the Swift–Hohenberg equation, the Ginzburg–Landau (or Allen–Cahn) equation and some model from surface growth. We discuss the impact of degenerate noise on the dominant behaviour, and see that additive noise has the potential to stabilize the dynamics of the dominant modes. Furthermore, we discuss higher order corrections to the amplitude equation.

Thin-film-growth models: roughness and correlation functions

Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of a solution and statistical quantities (e.g. mean interface width or correlation functions). Moreover, we discuss the approximations of such statistical quantities by the spectral Galerkin method. This is an important question, as the numerical computation of statistical quantities plays a key role in the verification of the models.

Front Motion in the One-Dimensional Stochastic Cahn--Hilliard Equation

In this paper, we consider the one-dimensional Cahn--Hilliard equation perturbed by additive noise and study the dynamics of interfaces for the stochastic model. The noise is smooth in space and defined as a Fourier series with independent Brownian motions in time. Motivated by the work of Bates and Xun on slow manifolds for the integrated Cahn--Hilliard equation, our analysis reveals the significant difficulties and differences in comparison to the deterministic problem. New higher order terms that we estimate appear due to Ito calculus and stochastic integration and dominate the exponentially slow deterministic dynamics. Using a local coordinate system and defining the admissible interface positions as a multidimensional diffusion process, we derive a first order linear system of stochastic ordinary differential equations approximating the equations of front motion. Furthermore, we prove stochastic stability of the approximate slow manifold of solutions over a very long time scale and evaluate the noise...