Prof. Dr. Dirk Blömker (Institut für Mathematik): Approximate slow manifolds for SPDEs
For a stochastic partial differential equation we approximate the infinite dimensional stochasticdynamics by the motion along a finite dimensional slow manifold. This manifold is deterministic,but not necessarily invariant for the dynamics of the unperturbedequation.
Our main results are the derivation of an effective equation (given bya stochastic ordinarydifferential equations) on the slow manifold,and furthermore the stochastic stability of the manifoldin the sense that with probability almost 1 the solution stay close tothe manifold for very long times.
We present applications to motion of multiple kinks for the stochasticone-dimensional Cahn-Hilliard equation,the motion of a single droplet along the boundary of the domain in thetwo- or three dimensional mass-conservative Allen-Cahn equation,and the motion of droplets in the stochastic Cahn-Hilliard equation.