Well-posedness And Blow-up Phenomena For Weakly Dissipative Shallow Water Wave Equations

In this doctoral dissertation, we study local well-posedness, precise blow-up scenario and blow-up phenomena, global existence of solutions to the Cauchy problems for a class of nonlinearly dispersive wave equations, which arise from the branches of modern mechanics and physics, for example:hydromechanics, elasticity and so on. The dissertation is divided into six chapters. Here we present the conclusions obtained in this dissertation. The specific content is as follows.In the first chapter, we first introduce the research background and the research significance of the equations. Then, we give the research status of the equations, and present the research ideas and conclusions of the dissertation. We finally introduce the conclusions of the dissertation.In the second chapter, we introduce the preliminary knowledge of the dissertation: some related definitions, theorems, inequalities and notations of this dissertation.In the third chapter, we mainly discuss the Cauchy problem for a weakly dissipative periodic two-component Hunter-Saxton system. We first establish the local well-posedness for the system by Kato theory. We then present a precise blow-up scenario of solutions to the system by the local analysis method of transport equations. We known that the blow-up phenomena of the system is concerned with the slope of the first component of the strong solutions of system, and do not depends on the slope of the second component of the strong solutions of system. Next, there are various detailed results of wave breaking and blow-up rate of solutions. We find a result:the blow-up rate do not depends on the weakly dissipative term. Finally, we provide a sufficient condition for global solutions by constructing Lyapunov function.In the fourth chapter, we study local well-posedness, precise blow-up scenario and blowup phenomena of solutions to the Cauchy problems for a weakly dissipative modified Camassa-Holm equation. We first show that the equation is locally well-posedness by Kato’s theorem. By the local analysis method of transport equations, detailed blow-up criteria for strong solutions are established. It is shown that the solutions of the modified CH equation can only have singularities which correspond to wave breaking. Finally, using estimation of inequalities, a sufficient condition for wave breaking of strong solutions in finite time is specified. In the fifth chapter, we study the local well-posedness, blowup phenomena and the global existence result for a weakly dissipative higher-order Camassa-Holm equation. Firstly, by applying the Kato theory, the local well-posedness of the higher-order Camassa-Holm equation is studied. Next, by the weakly dissipative term, we obtain the blow-up result for the strong solution without conservation law. Finally, by applying this norm estimate, the global existence result is obtained.In the last chapter, we present the conclusions of this dissertation and the prospect of future work.