We prove the existence and uniqueness of pulsating waves for the motion by mean curvature of an n-dimensional hypersurface in an inhomogeneous medium, represented by a periodic forcing. The main difficulty is caused by the degeneracy of the equation and the fact the forcing is allowed to change sign. Under the assumption of weak inhomogeneity, we obtain uniform oscillation and gradient bounds so that the evolving surface can be written as a graph over a reference hyperplane. The existence of an effective speed of propagation is established for any normal direction. We further prove the Lipschitz continuity of the speed with respect to the normal and various stability properties of the pulsating wave. The results are related to the homogenisation of mean curvature flow with forcing.

Abstract. The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption-spin flip mechanisms in the context of surface processes. For such an equation we consider a multiplicative space-time white noise with diffusion coefficient of sub-linear growth. Using technics from semigroup theory, we prove existence, and path regularity of stochastic solution depending on that of the initial condition. Our results are also valid for the stochastic Cahn-Hilliard equation with unbounded noise diffusion, for which previous results were established only in the framework of a bounded diffusion coefficient. We prove that the path regularity of stochastic solution depends on that of the initial condition, and are identical to those proved for the stochastic Cahn-Hilliard equation and a bounded noise diffusion coefficient. If the initial condition vanishes, they are strictly less than 2 − d 2 in space and 1 2 − d 8 in time. As expected from the theory of parabolic operators in the sense of Petrovskȋı, the bi-Laplacian operator seems to be dominant in the combined model.

We study the ε-dependent two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit ε → 0. The parameter ε is positive and measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling, our results indicate that the stochastic Cahn-Hilliard equation converge to a Hele-Shaw problem with stochastic forcing on the curvature equation. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.Résumé: Nousétudions l'équation Cahn-Hilliard stochastique dépendante en ε, posée en dimensions deux et trois dans la limite de l'interface nette ε → 0. Le paramètre ε est positif et mesure la largeur de couches de transition générées pendant la séparation de phase. Nous avons aussi couplé la puissance de bruità ce paramètre.À l'aide de séries asymptotiques formelles, nous déterminons la limite. Dans l'échelle propre, nos résultats indiquent que l'équation Cahn-Hilliard stochastique converge vers un problème Hele-Shaw avec une force stochastique présenteà l'équation de la courbure. En cas de bruit suffisamment petit, nous prouvons rigoureusement que la limite est un problème Hele-Shaw déterministe. Finalement, nous discutons quelles estimations sont nécessaires afin de prolonger le résultat rigoureux en présence de bruit d'une puissance plus grande.Here u : D × [0, T ] → R is the scalar concentration field of one of the components in a separation process, for example of binary alloys.

Abstract. We study the stochastic mass-conserving Allen-Cahn equation posed on a smoothly bounded domain of R 2 with additive, spatially smooth, space-time noise. This equation describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point of locally maximum curvature. We apply Itô calculus to derive the stochastic dynamics of the center of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco [2] for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a sufficiently small noise strength, we establish stochastic stability of a neighborhood of the manifold of boundary droplet states in the L 2 -and H 1 -norms, which means that with overwhelming probability the solution stays close to the manifold for very long time-scales.