Attractive invariant manifolds under approximation. Inertial manifolds

A class of nonlinear dissipative partial difTcrential equations that possess finite dimensional attractive invariant manifolds is considered An existence and perturba-tion theory is developed which unifies the cases of unstable manifolds and inertialmanifolds into a single framework. It is shown that certain approximations of theseequations, such as those arising from spectral or finite element methods in space, one-step lime-discreti/ation or a combination of both, also have attractive invariantmanifolds. Convergence of the approximate manifolds to the true manifolds isestablished as the approximation is refined. In this part of the paper applicationsto the behavior of inertial manifolds under approximation are considered. Fromthis analysis deductions about the structure of the attractor and the flow on theattractor under discretization can be made.