Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions

[摘要] We prove that a general class of nonlinear, non-autonomous ODEs in Frechet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ODE are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. The main motivation is the case where the Frechet space in question is a suitable function space, and the maps involved in an ODE in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ODE is a finite-dimensional PDE. We show that our methods apply to a relevant class of nonlinear, non-autonomous PDEs in this way. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-05

[关键词] Nonlinear;Non-autonomous PDE;Invariant manifold;Normal form;Differentiable and polynomial maps between Frechet spaces