On the global behavior of the solutions of the Navier-Stokes equations and related PDEs

We study two different features of the long term behavior of the Navier-Stokes equations. In the first part of the dissertation, we introduce a new method for constructing approximate inertial manifolds for 2D Navier-Stokes equations. Namely, we slightly modify the spectrum of the dissipative linear operator of the Navier-Stokes equations such that the resulting equation possesses an inertial manifold. We conclude that the inertial manifold for the modified equation can be regarded as an approximate inertial manifold for the Navier-Stokes equation by proving that the inertial manifold for the modified equation is close to the attractor of Navier-Stokes equations. We also demonstrate how this technique can be applied to other related PDEs such as Kuramoto-Sivashinsky Equations. The second part of the dissertation is devoted to the study of the non analytic Gevery class regularity for weak solutions of 3D Navier-Stokes equations and a global estimate is obtained for all weak solutions.