Modulation- and Amplitude-Equations for stochastic partial differential equations

Vortrag von Prof. Dr. Dirk Blömker

Datum: 11.10.17  Zeit: 16.15 - 17.15  Raum: ETH HG E 1.2

Modulation- or Amplitude-Equations are a universal tool to approximate solutions of complicated systems given by partial or stochastic partial differential equations (SPDEs) near a change of stability, when there is no center manifold theory available. Relying on the natural separation of time-scales at the bifurcation the solution of the original equation is well approximated by the bifurcating pattern. For the talk we consider for simplicity the one-dimensional stochastic Swift-Hohenberg equation, which acts as a toy model for the convective instability in Rayleigh-Benard convection. On an unbounded spatial domain the amplitude of the dominating pattern is slowly modulated in time and also in space. Furthermore, it solves a stochastic Ginzburg-Landau equation perturbed by an additive space-time white noise. Major problems arise due to the weak regularity of solutions and their unboundedness in space, so that the methods from the theory of deterministic modulation equations all fail.