Topological Stability Of The Hyperbolic Compact Invariant Manifolds In The Infnite Dimensional Space

Our goal in this paper is to study the persistence for perturbations of hyperbolic, compact, invariant manifolds in an infnite dimensional setting. Let A be a positive sectorial operator on a Banach space W, M be a compact hyperbolic invariant manifold for a semigroup S1(t) generated by a given evolutionary equation δtu+Au=F(u) on a Banach space W We prove that the topological property of S1(t) on M, that is, for every ε>0, there is a δ>0, such that if‖G‖{A;C1(Ω)}<δ, then there is a continuous mapping h:Mâ†'W and a strictly increasing mapping φ:R+â†'R+, one has‖Aβ(h-I)‖<2ε, and for the semigroup S2G(t) generated by the evolutionary equation δty+Ay=F(y)+G(y), with the property that h°S1(φ(t))=S2G(t)°h on M. In order to prove this result, we needs the nonlinear terms F and G are belong to the space CLIP1, whereOur result can be easily applied in wide range of applications, including the Navier-Stokes equations. Of special interest in this paper is the role that this theory of persistence of invariant manifolds plays in the context of the numerical analysis of the longtime dynamics of solutions of partial differential equations. It is noteworthy in this regard that, in order to be able to apply this result in the analysis of numerical schemes used to study discretizations of partial differential equations, we need use a new norm of the perturbation term G defined by Where L=L(V2β,W).