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Amplitude equation for SPDEs with quadratic nonlinearities. (English) Zbl 1197.60061

Summary: We rigorously derive stochastic amplitude equations for a rather general class of SPDEs with quadratic nonlinearities forced by small additive noise. Near a change of stability, we use the natural separation of time-scales to show that the solution of the original SPDE is approximated by the solution of an amplitude equation, which describes the evolution of dominant modes. Our results significantly improve older results. We focus on equations with quadratic nonlinearities and give applications to the one-dimensional Burgers equation and a model from surface growth.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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