Hyperdissipative Navier-Stokes Equations Driven by Time-Dependent Forces: Invariant Manifolds

We consider the incompressible hyperdissipative Navier-Stokes equations \biggl\{ ut + \epsilon(-\Delta )\alphau + (u \cdot O)u + Op = f, O\cdotu = 0, on a two-or three-dimensional periodic torus, where the power \alpha \geq 3/2 and the forcing function f is time-dependent. We intend to reveal how the fractional dissipation and the time-dependent force affect long-time dynamics of weak solutions. More precisely, we prove that with certain conditions on f, there exists a finite-dimensional Lipschitz manifold in the L2-space of divergent-free vector fields with zero mean. The manifold is locally forward invariant and pullback exponentially attracting. Moreover, the compact uniform attractor is contained in the union of all fibers of the manifold. In our result, no large viscosity \epsilon is assumed. It is also significant that in the three-dimensional case the spectrum of the fractional Laplacian (-\Delta )3/2 does not have arbitrarily large gaps.