Interface motion for the stochastic Allen-Cahn and Cahn-Hilliard equation

In this thesis, we consider the stochastic Cahn-Hilliard-Cook and (mass conserving) Allen-Cahn equation in the physically relevant space dimensions. Both of these equations serve as phenomenological model for the phase separation and subsequent coarsening of a two-component mixture. In our studies, we focus on the almost final stage of the evolution, when after an initial spinodal decomposition or nucleation the mixture is well-separated, and the dynamics is given by the motion of an interface on a metastable slow manifold. In the one-dimensional setting, the slow manifold is parametrized by the zeros of a profile having a finite number of transitions from one pure phase into the other. In higher space dimensions for very late stages of separation, the transition between phases occurs in a small neighborhood of an almost spherical interface. Here, the metastable manifold consists of translations of a droplet state with a fixed size. We derive the effective equation on the slow manifold via an orthogonal projection for a relatively small noise and small atomistic interaction length. Thus, the underlying infinite-dimensional system can be described to very high accuracy by a finite-dimensional stochastic ordinary differential equation. We will see that the thermal fluctuations dominate the dynamics. This is quite different to the deterministic case, where at this stage the evolution is exponentially slow in the atomistic interaction length. We analyze the stochastic stability and show that solutions stay close to the slow manifold for a very long time with high probability. Crucial for the stability analysis are spectral estimates of the linearization around the energetically favorable states