The Asymptotic Behaviors For Several Equations Of Fluid Dynamics

Infinite dimensional dynamical systems is a subject with wide applications in various fields. It mainly focuses on the global existence and the asymptotic behavior of the solutions to some nonlinear dissipative evolutionary equations, which arise in physics, chemistry, fluid dynamics, biology, atmospheric science et al. The study of the asymptotic behavior of these dissipative systems is of great theoretical and practical importance, as it is essential, for practical applications, to be able to understand, and predict the long time behavior of the solutions of such systems. In this doctoral dissertation, we consider the global existence and the long-time behavior of solutions for some evolutionary equations of fluid dynamics from the point view of infinite dynamical systems.This dissertation is divided into six chapters. In Chapter one, we sketch some basic problems and frontiers of the infinite dimensional dynamic systems. We will mainly focus on the theory of global and exponential attractors for the autonomous dynamical systems, uniform and pullback attractors for non autonomous dynamical systems, and the dimen-sion of the attractors. In Chapter two, we provide some notations of the function spaces and some useful inequalities involved in this dissertation.In Chapter three, we consider the stability of the third grade fluid equations in the whole space IR3. We prove a global stability result and an asymptotic stability result, which improve the existence result in the literature.In Chapter four, we study the asymptotic behavior of solutions to the non autonomous third grade fluid equations with spacial periodic boundary conditions on bounded domains. We first prove the existence of a uniform attractor to the solution process in proper spaces. Then we explore the convergence of solutions and the uniform attractor to those of Navier-Stokes equations. In the case of dimension two, we prove that the weak solution and the uniform attractor of the third grade fluid equations converge, respectively, to the weak solution and the uniform attractor of the 2d Navier-Stokes equations as the parameters α,β tend to zero.In Chapter Five, we consider the large time behavior of the third grade fluid type MHD system, which is given by the coupling of a class of third grade fluid equation and the Maxwell equation. We first prove the existence and uniqueness of a global solution. Then using the short trajectory method, we prove that the system admits a finite dimensional global attractor and an exponential attractor.In Chapter six, we investigate the long time behavior of solutions to the following generalized Navier-Stokes equations. where 3/4<α<1,Fn(r)=min{1,N/r},(?r ∈IR+. Under suitable assumptions on the initial data and forcing terms, we prove the existence and uniqueness of a global solution. Moreover, we prove that the solution semigroup admits a unique global attractor. The upper bound for the fractal dimension of the attractor is also provided.