Existence of inertial manifolds for the Navier–Stokes equations

Establish the existence of inertial manifolds for the Navier–Stokes equations in the general setting, beyond the specific periodic-solution and spherical-domain framework considered in the paper.

Background

The paper studies Navier–Stokes equations for divergence-free vector fields on a d-dimensional sphere under a periodic external force. It proves the existence and uniqueness of a periodic mild solution for sufficiently small forcing and constructs an inertial manifold for a cut-off perturbation equation around that solution when an appropriate spectral-gap condition holds.

Despite these results in a restricted geometric and dynamical setting, the general existence of inertial manifolds for the Navier–Stokes equations is identified as unresolved. The problem is significant because inertial manifolds would provide finite-dimensional reductions capturing the long-time dynamics of an infinite-dimensional fluid system.