The Behaviors Of The Solution Of Vlasov -Darwin(Nordstr(?)m) System And The Two-component Camassa-Holm System

The Ph.D thesis is devoted to the investigation on the behaviour of the solution to several nonlinear evolution equations in shallow water wave theory and plasma theory, es-pecially the well-posedness problem of solution. It is divided into two parts:one part focus on the behaviour of solutions to the Vlasov equation in plasma theory; the other part con-sider the local well-posedness in the framework of Besov space, blow-up criteria and the sufficient condition for global strong solution to the initial value problem of the generalized Camassa-Holm equation in the shallow water wave theory.The first chapter introduce the background information, research progress on the sub-ject of this paper, and state the main results of this thesis.The second chapter is devoted to the study on the well-posedness of Relativistic Vlasov Darwin system. In the three dimensional case, Firstly, we obtain the propagation of regular-ity for the relativistic Vlasov Darwin system with sufficient smooth initial value; Secondly, because global existence and uniqueness of classical solutions to the Cauchy problem is an open problem, under small perturbations of the initial data, the behaviour of classical solu-tions to the system is considered. Based on the conclusion, the global existence of classical solutions for nearly spherically symmetric initial value is obtained.The third chapter focus on the energy conservation law of weak solution to 3D Vlasov-Nordstrom-system. The field equation in Vlasov-Nordstrom is given by Nordstrom scalar field theory. Because of the complexity of the Vlasov-Einstein model,it is hoped that by studying the Vlasov-Nordstrom system one can learn more about the Vlasov-Einstein model or other kinetics equations. In 2004, S.Calogero and G.Rein obtain the global existence of weak solution to Cauchy problem, and show that the total energy is bounded by the initial value. Under additional conditions, we obtain the conservation law of total energy for the weak solutions by the regularization method.In the fourth chapter, we consider the Cauchy problem and blow-up criteria for a new integrable two-component Camassa-Holm system. The local well-posedness in Besov space and the blow-up criteria for strong solution to the initial value problem are studied In[130]. Further the first result in the chapter discuss the local well-posedness of solution to the system in critical Besov space .by applying Littlewood-Paley decomposition theory and a prior estimate of transport equation, and then under certain initial conditions, the existence of global solution in some Sobolev space is established. Finally with the help of the commutator estimate and conservation law, we establish the blow-up criteria in finite time in critical Besov space.The fifth chapter study the Cauchy problem and blow-up criteria of a new integrable two-component Camassa-Hlm system. Firstly, local well-posedness of the solution of this equation in Besov space is proved, and by means of the log-type interpolation inequality and Osgood lemma, We also get the local well-posedness of solution in critical Besov space. In addition, with some conditions on the initial value, two kinds of blow-up phenomena of the equation are established.Finally, we shall summarize the main work of the Ph.D thesis and outline some prospects about it.