Lecture Notes: Modulation- and Amplitude-Equations for stochastic partial differential equations
نویسنده
چکیده
Modulationor 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 with an amplitude that is slowly modulated in time and also in space, if the underlying domain is sufficiently large. The approach is on a formal level well known in the physics literature, and for partial differential equations on unbounded domains rigorously studied in the last two decades in numerous publications. For the course we focus on rigorous results for SPDEs. We address first the case of bounded domains, where the amplitude is slowly moving in time and satisfies a (stochastic) ordinary differential equation. This is a qualitative replacement for center manifold theory and we discuss approximation of invariant measures and pattern formation below change of stability. In a second part, we focus on results for SPDEs on sufficiently large or unbounded domains, where the amplitude of the domination pattern is also slowly modulated in space and solves a SPDE. We discuss the impact of the size of the underlying domain, stabilization due to degenerate additive noise. Finally, we address problems with SPDEs on unbounded domains with additive noise. For simplicity of presentation and to avoid technicalities we consider mostly the stochastic Swift-Hohenberg equation, which acts as a toy model for the convective instability in Rayleigh-Benard convection. These notes should summarize in a non-technical and partially non-rigorous way the main ideas and results presented in the two lectures of the minicourse “Modulationand Amplitude-Equations for stochastic partial differential equations” and the colloquium talk on July 15th 2015.
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