Certain Partial Differential Equations And Its Infinite Dimensional Dynamical System In The Geophysical Fluid Dynamics

The dissertation is devoted to considering the primitive equations of the atmosphere and the oceans and their corresponding infinite-dimensional dynamical systems in geophysical fliud dynamics. Moreover, we consider a dissipative two-dimensional quasi-geostrophic equation under random forcing. Now the subject, geophysical fliud dynamics, mainly tends to focus on the dynamics of large-scale phenomena in the atmosphere and the oceans. One of the important contents in the dynamics is to study the infinite-dimensional dynamical systems of the atmosphere and the oceans. The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to attack the problem is to consider its global attractors. We prove rigorously in mathematics the existence of global attractors of the atmosphere and the oceans. We also obtain the existence of random attractor for the stochastic quasi-geostrophic dynamical system. Our results play some roles in the fundamental study of the motion of the atmosphere and the oceans. The dissertation consists of four chapters.In chapter 1, we briefly introduce the background in physics and developments of the primitive equations of the atmosphere and the oceans and their corresponding infinite-dimensional dynamical systems. In addition, the main of the dissertation is described.In Chapter 2, we consider the initial boundary value problem for the primitive equations of the moist atmosphere which are used to describe the turbulent behavior of long-term weather prediction and climate changes. By Faedo-Galerkin method and compactness theories, we obtain the existence of global weak solutions to the problem in large-scale atmosphere. By studying the longtime behavior of solutions, we obtain trajectory and global attractors for the infinite-dimensional dynamical system of the moist atmosphere.In Chapter 3, we consider the initial boundary value problem for the three-dimensional viscous primitive equations of large-scale moist atmosphere. First, we obtain the existence and uniqueness of global strong solutions of the problem. Second, by studying the long-time behavior of strong solutions, we construct a universal attractor A which captures all the trajectories. In Chapter 4, we consider a dissipative two-dimensional quasi-geostrophic equation, which model a class of large-scale geophysical flows under random forcing (the forcing is a Gaussian random field, white noise in time). First, we prove the existence and uniqueness of the global solution to the initial boundary value problem for the stochastic equation. Second, by studying the asymptotic behavior of the solution, we obtain the existence of random attractor for the stochastic quasi-geostrophic dynamical system.