On the Solutions in the Global Attractor of the Incompressible Navier-Stokes Equations

Abstract

We study the global attractor for the solutions of the incompressible Navier-Stokes equations (NSE) equipped with appropriate boundary conditions.

A challenge in the cases when zero is not in the global attractor is to find sharp lower bound on the energy. A related challenging problem is to show that zero is in the attractor if and only if the external force is zero. We show that if zero were in the global attractor, then all its elements, as well as the external force, must be smooth functions. By exploring a particular family of function classes, we show that the set of nonzero external forces for which zero could be in the global attractor is meagre (of the first Baire category in a Fréchet topology).

The weak global attractor of three dimensional Navier-Stokes equations is a complex geometric object. An interesting challenging question is to measure its complexity. Invoking the fact that topology on the weak global attractor can be metrizable, we use a physically reasonable metric function to obtain explicit estimate for the Kolmogorov e-entropy of the weak global attractor in terms of the physical parameter associated with the fluid flow.

We also study the existence of the nonstationary solutions in the global attractor of the space periodic two dimensional NSE which have constant energy and enstropy per unit mass for all time. A subclass of such solutions whose geometric structures have a supplementary stability property is defined and explored. We prove that the wave vectors of the active mode of this subclass must satisfy a finite Galerkin system. The nonexistence of solutions in this subclass is proved for the particular case when the external force has a special property.