Perturbations of foliated bundles and evolutionary equations

Abstract

In two earlier papers, we presented a perturbation theory for laminated, or foliated, invariant sets \(\mathcal{K}^o\) for a given finite-dimensional system of ordinary differential equations, see [20,21]. The main objective in that perturbation theory is to show that: if the given vector field has a suitable exponential trichotomy on \(\mathcal{K}^o\), then any perturbed system that is C1-close to the given vector field near \(\mathcal{K}^o\) has an invariant set \(\mathcal{K}^n\), where \(\mathcal{K}^n\) is homeomorphic to \(\mathcal{K}^o\) and where the perturbed vector field has an exponential trichotomy on \(\mathcal{K}^n\).In this paper we present a dual-faceted extension of this perturbation theory to include: (1) a class of infinite-dimensional evolutionary equations that arise in the study of reaction diffusion equations and the Navier–Stokes equations and (2) nonautonomous evolutionary equations in both finite and infinite dimensions. For the nonautonomous problem, we require that the time-dependent terms in the problem lie in a compact, invariant set M. For example, M may be the hull of an almost periodic, or a quasiperiodic, function of time.