Time dependent center manifold in PDEs

Hongyu Cheng; Rafael de la Llave

doi:10.3934/dcds.2020213

Article Contents

2020, Volume 40, Issue 12: 6709-6745. Doi: 10.3934/dcds.2020213

This issue Previous Article Computer assisted proofs of two-dimensional attracting invariant tori for ODEs Next Article On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Time dependent center manifold in PDEs

Hongyu Cheng^1,2, and
Rafael de la Llave^2,

1.
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
2.
School of Mathematics, Georgia Inst. of Technology, Atlanta GA, 30332, USA

Received: January 2019

Revised: February 2020

Early access: May 2020

Published: August 2020

H. C. supported by CSC by the National Natural Science Foundation of China (Grant Nos. 11171185, 10871117), H.C. thanks G.T. for hospitality 2015-2016.
R. L. supported in part by NSF grant DMS 1800241.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester.

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Abstract

We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution).

We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a "time-dependent invariant manifold" (TDIM). These manifolds evolve with the forcing. They are such that the original equation is always tangent to a vector field in the manifold. Hence, for initial data in the TDIM, the original equation is equivalent to an ordinary differential equation. This allows us to define families of solutions of the full equation by studying the solutions of a finite dimensional system. Note that this strategy may apply even if the original equation is ill posed and does not admit solutions for arbitrary initial conditions (the TDIM selects initial conditions for which solutions exist). It also allows that the TDIM is infinite dimensional.

Secondly, we construct the center manifold for skew systems driven by the external forcing.

Thirdly, we present concrete applications of the abstract result to the differential equations whose linear operators are exponential trichotomy subject to quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.

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Mathematics Subject Classification: 35R25, 37L10, 35J60, 47J06, 37L25.

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