Attractive Invariant Manifolds under Approximation. Inertial Manifolds

Abstract

A class of nonlinear dissipative partial differential equations that possess finite dimensional attractive invariant manifolds is considered. An existence and perturbation theory is developed which unifies the cases of unstable manifolds and inertial manifolds into a single framework. It is shown that certain approximations of these equations, such as those arising from spectral or finite element methods in space, one-step time-discretization or a combination of both. also have attractive invariant manifolds. Convergence of the approximate manifolds to the true manifolds is established as the approximation is refined. In this part of the paper applications to the behavior of inertial manifolds under approximation are considered. From this analysis deductions about the structure of the attractor and the flow on the attractor under discretization can be made.