Center Manifolds for Rough Partial Differential Equations

Christian Kuehn; Alexandra Neamtu

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arXiv · 2111.01488

Center Manifolds for Rough Partial Differential Equations

Christian Kuehn·Alexandra Neamtu

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Abstract

We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a $\gamma$-H\"older rough path, for $\gamma\in(1/3,1/2]$. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical systems (RDS). We also illustrate our main theorem for reaction-diffusion equations as well as for the Swift-Hohenberg equation.

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BibTeX RIS

Christian Kuehn, Alexandra Neamtu. 2021-11-02. Center Manifolds for Rough Partial Differential Equations. https://arxiv.org/abs/2111.01488

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