Floquet Problem and Center Manifold Reduction for Ordinary Differential Operators with Periodic Coefficients in Hilbert Spaces

Vladimir Kozlov; Jari Taskinen

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arXiv · 1905.07890

Floquet Problem and Center Manifold Reduction for Ordinary Differential Operators with Periodic Coefficients in Hilbert Spaces

Vladimir Kozlov·Jari Taskinen

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Abstract

A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite dimensional system of ordinary differential equations with constant coefficients and an infinite dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders (Kuchment) and for elliptic problems in quasicylinders (Nazarov). As an application we give a center manifold reduction for a class of non-linear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients (Mielke).

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Vladimir Kozlov, Jari Taskinen. 2019-05-20. Floquet Problem and Center Manifold Reduction for Ordinary Differential Operators with Periodic Coefficients in Hilbert Spaces. https://arxiv.org/abs/1905.07890

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