One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters – finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems to design finite-dimensional feedback control for stabilizing their solutions. Notably, it is observed that this very same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete measurements. In addition, I will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named determining form, which is governed by a globally Lipschitz vector field. Remarkably, as a result of this machinery I will eventually show that the global dynamics of the Navier-Stokes equations is being determining by only one parameter that is governed by an ODE. The Navier-Stokes equations are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative reaction-diffusion systems and geophysical models.[-]

In this course I will be covering three main topics. The first part will be concerning the NavierStokes and Euler equations - a quick survey. The second part will discuss the question of global regularity of certain geophysical flows. The third part about coupling the atmospheric models with the microphysics dynamics of moisture in warm clouds formation.
The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which are called the 'Primitive Equations', is often prohibitively expensive computationally, and hard to study analytically. In these introductory lectures, aimed toward graduate students and postdocs, I will survey the mathematical theory of the 2D and 3D Navier-Stokes and Euler equations, and stress the main obstacles in proving the global regularity for the 3D case, and the computational challenge in their direct numerical simulations. In addition, I will emphasize the issues facing the turbulence community in their turbulence closure models. However, taking advantage of certain geophysical balances and situations, such as geostrophic balance and the shallowness of the ocean and atmosphere, I will show how geophysicists derive more simplified models which are easier to study analytically. In particular, I will prove the global regularity for 3D planetary geophysical models and the Primitive equations of large scale oceanic and atmospheric dynamics with various kinds of anisotropic viscosity and diffusion. Moreover, I will also show that for certain class of initial data the solutions of the inviscid 2D and 3D Primitive Equations blowup (develop a singularity).[-]