Global regularity for certain dissipative hydrodynamical and geophysical systems with an application in control theory

In this dissertation, we deal with different properties of the solutions for several dissipative evolution systems. In one we study the regularity, namely, a Gevrey class regularity of the solution for the nonlinear analytic parabolic equations and Navier-Stokes equations on the two dimensional sphere. We prove the instantaneous Gevrey regularity for these systems. In addition, we provide an estimate for the number of determining modes and nodes for the two dimensional turbulent flows on the sphere. Next, we study the existence and uniqueness of the Lake equations, a special shallow water model of a fluid flow in a shallow basin with varying bottom topography. We show that the global existence of weak solutions for these equations with certain degenerate varying bottom topography, i.e., in the presence of beaches. Later we show the uniqueness for the case of non-degenerate but non-regular topography. Finally, we consider a feedback control problem for the Navier-Stokes equations. Namely, we show that in case one is able to design a linear feedback control that stabilizes a stationary solution to the Galerkin approximating scheme of the Navier-Stokes equations then the same feedback controller is, in fact, stabilizing a near by exact steady state of the closed-loop Navier-Stokes equations. It is worth to stressing that all the conditions of this statement are checkable on the computed Galerkin approximating solution. The same result is also true in the context of nonlinear Galerkin methods, which based on the theory of Approximate Inertial Manifolds, and for various other nonlinear dissipative parabolic systems.