The Long Time Behavior Of The Nonlinear Boussinesq Flow

This dissertation consists of two parts. In one part we consider the ex- istence of the time periodic solutions, exponential attractors and the upper semi-continuity of the global attractors for the coupled system of equations of fluid and temperature in the Boussinesq approximation. In the other part, we consider the existence of the time periodic solutions and regularity of the global attractors for the coupled system NLS-Boussinesq. This dissertation consists of six chapters. In chapter 1, we briefly in- troduce background in physics and the developments in mathematics for the coupled system of equations of fluid and temperature in the Boussinesq ap- proximation and the coupled system NLS-Boussinesq. in which the main work of the dissertation is also described. In chapter 2, we discuss the time peri- odic solution of the coupled system of equations of fluid and temperature in Boussinesq approximation. First, we apply the fixed theorem of the Larey- Schauder to prove the existence of the approximate solution. Next, we get the estimates of the higher order derivatives (with respect to spatial variable and time variable) of the approximate solutions. Finally, we use the method of standard compactness arguments to get the existence of this system in a bounded domain ~l (€ Rd, d 3,4), whenever the external force are small. At the same time, after supplementing some conditions we get the result of uniqueness. In chapter 3, we study the existence of the exponential attractors for the coupled system of equations of fluid and temperature in Boussinesq approximation. We first show that the solution operator S(t) is Lipschitz con- tinuous, then the discrete solution operator S~ (t*) satisfy the squeezing property, use the theory given by A. Eden and C. Foias, we get the existence of the exponential attractors M whose fractal dimension is finite. In chapter 4, we study the upper semi-continuity of global attractors for the coupled system of equations fluid and temperature in Boussinesq approximation. Considering the equations with the singularly pertubed term, and decomposing the solu- tion operator, we first prove the existence of the global attractors .4(&), then prove that lim dist(A(5),A(O)) = 0. ?0+ In chapter 5, using the same arguments in chapter 2 for the coupled system of NLS-Boussinesq, we prove the existence of the time periodic solution and point out that the time periodic solution is unique when the external force are small. In chapter 6, using the technology of decomposing solution operator and constructing the asymptotic compact invariant set, we get the existence of the global attractor ..4~ in the space E0. Furthermore, A0 is the global attractor in the space E0, that is A0 = A0.