A Remark on Quasi-Stationary Approximate Inertial Manifolds for the Navier–Stokes Equations

Abstract

The two dimensional Navier–Stokes equations with time-dependent external body forces is considered. Under appropriate assumptions on the temporal properties of the forcing term the authors are able to construct a time-dependent deterministic approximate inertial manifold. It is shown that all solutions converge exponentially fast to a thin neighborhood of this manifold. If the forcing term is too oscillatory in time, it is shown by example that the techniques used in the construction of certain approximate inertial manifolds for the autonomous case, in general, do not extend to the time-dependent case. Also it is shown that if the forcing term is time-independent and spatially smooth (Gevrey class), then the global attractor lies exponentially close to the linear manifold spanned by the first m eigenfunctions of the Stokes operator, provided m is large enough.