| Abstract: |
| We consider the approximation via modulation equations for nonlinear stochastic partial differential equations (SPDEs) like the stochastic Swift-Hohenberg (SH) equation on the unbounded domain with additive space time white noise. Close to a bifurcation of a single mode a small band of infinitely many eigenvalues changes stability. Thus solutions of SH are well described by a modulated wave, where the amplitude solves a stochastic Ginzburg-Landau (GL) equation with space time white noise.
In the one-dimensional case due to the weak regularity of solutions the standard deterministic methods for modulation equations fail, as we need weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise. Moreover, solutions of GL are only H\older continuous, which gives just enough regularity to obtain the approximation result.
In the two-dimensional case one runs into the problem that solutions of GL with space-time white noise are no longer well defined. |
|