The Study Of Fluid System Based On The Littlewood-Paley Theory

Control theory and technology are widely used in many fields in our real lives and they play the more and more important roles. The study of control problems for dis-tributed parameter systems is very active. In fact, many distributed parameter systems are described by the partial differential equations. The mathematical physical models described by partial differential equations may reflect the nature of natural phenomenon. As one of the fundamental equations in Mathematics and Physics, the equation which models the propagation of fluid has strong practical application backgrounds. For exam-ple, the application in fluid mechanics, elastic mechanics and control theory. Nonlinearity is a common phenomenon in the nature and engineering technology fields. For the con-trol problems of nonlinear partial differential equations, we expect that the problems are well-posed at first. Recently, Fourier analysis has been widely used in the study of partial differential equations. In particular, the Littlewood-Paley decomposition and Bony paraproduct decomposition method are effective tools. This thesis is devoted to using the Littlewood-Paley theory to investigate the well-posedness for Cauchy prob-lems of the Camassa-Holm system, Degasperis-Procesi system, liquid crystal system in the Besov spaces and properties of solutions in the Sobolev spaces. We also consider the optimal control problem for shallow water equation with a viscous term and the global stabilization of Camassa-Holm equation with a distributed feedback control. The itera-tive learning control of generalized Korteweg-de Vries equation with a damping term is studied. This thesis is concerned with the stabilization of three kinds of fluid systems with strong physical and engineering backgrounds and the related control problems. We achieve some interesting results.The weakly dissipative Camassa-Holm equation, Camassa-Holm system and dis-sipative Camassa-Holm system are studied. The local well-posedness for the Cauchy problems are established by using the Littlewood-Paley theory and priori estimates of solutions to transport equation. We also analyze the blow-up phenomenon and blow-up rate of solutions. The global existence of solutions is obtained by constructing a Lya-punov function. In particular, the infinite propagation speed of solutions to the dissipa-tive Camassa-Holm system is investigated. The relation between dissipation coefficient ? and blow-up criterion, blow-up rate of solutions, the effects of dissipation coefficient ? and diffusion coefficient k on infinite propagation speed of solutions are given, respec-tively. The global stabilization of the Camassa-Holm equation with a distributed linear feedback control is studied. The global existence, uniqueness and blow-up criterion of strong solutions are obtained. The global existence, uniqueness and asymptotical stabi-lization of weak solutions are also established. It shows that the coefficients in feedback control are related to the exponential asymptotical stabilization of weak solutions.The dissipative Degasperis-Procesi system is investigated. Firstly, the well-posedness for the system with periodic boundary condition is established. Due to that the Degasperis-Procesi system does not possess the conservation law as Camassa-Holm system, we only obtain the blow-up criterion of solutions. In addition, the persistence property of solu-tions to the dissipative Degasperis-Procesi system is analyzed.The liquid crystal system is studied. By using the Littlewood-Paley decomposition and Bony decomposition method, we establish local well-posedness for the Cauchy prob-lem of liquid crystal system in the critical Besov spaces with negative index. Applying the contraction mapping principle, we obtain global well-posedness for the problem with small initial value. Thus, the regularity of solution spaces is improved as negative index. The blow-up criterion of solutions is given.The optimal control problem for a shallow water equation with a viscous term is in-vestigated. The existence of optimal control and optimal solution for the control problem are obtained by using the Galerkin method and optimal control theory of distributed pa-rameter systems. Using the Gateaux derivative of cost functional and adjoint equation, we obtain the first order necessity condition and local uniqueness of optimal control.The iterative learning control problem for the generalized Korteweg-de Vries equa-tion with a damping term is analyzed. The expression of states to the control system is established by using the semigroup theory. The priori estimates of states are obtained. The convergence condition of tracking errors is presented based on the P type iterative learning control algorithm, where it is permitted to that there exists some errors be-tween the initial value of states in iterative process. An numerical example illustrates the effectiveness of the proposed method.