Theory Of Invariant Manifolds For Nonautonomous Infinite-dimensional Dynamical Systems And Applications

This paper is devoted to exploiting a theory of finite-dimensional global manifold for nonautonomous infinite-dimensional dynamical system as well as its applications.We first consider an abstract nonautonomous dynamical system defined on a general Banach space.By adapting the classical graph transform method due to Hadamard,we prove that if a geometrical assumption,called local strong squeezing property,and several technical assumptions,called controllability,inverse Lipschitz property,and(partial)compactness property,are satisfied,then the system admits a finite-dimensional Lipschitz invariant manifold with an exponential tracking property acting on a local range and continuity.The above assumptions are independent of specific evolution equations,and do not involve any spectral gap condition.Subsequently,we apply this general framework to two types of nonautonomous evolution equations: Reaction-diffusion equations and Fitz Hugh-Nagumo systems.By modifying the nonlinearities of equations and using the classical principle of spatial averaging,we verify that the modified problems satisfy the aforementioned general assumptions.This yields the existence of finite-dimensional global manifolds.More precisely,the manifold is a family of submanifolds in the phase space.Each submanifold,having certain regularity of Sobolev type,can be represented as the graph of a Lipschitz map.Moreover,the manifold is locally forward invariant and exponentially pullback attracting,with respect to the corresponding nonautonomous dynamical system.If also the symbol space is compact,then the uniform attractor is contained in the union of all submanifolds.Finally,we consider the incompressible hyperdissipative Navier-Stokes equations under nonautonomous perturbations.Since the convection term differs essentially,in structure,from the nonlinearities of above two equations,another approach to modify the convection term is necessary.In addition,we introduce a new principle of spatial averaging which runs in the distribution sense,and then demonstrate that the modified problem verifies the strong squeezing property in the global sense.With this,the existence of finite-dimensional global manifold is proved.The linear unbounded operators in the principal part of these equations do not necessarily have arbitrarily large spectral gaps,meanwhile these equations are driven by timedependent additive forces.