Global Attractor For A Class Of Viscous Two-component Shallow Water Wave Equations

In this paper, we aim at considering the existence of global solution, absorbing set, and the global attractor for viscous two-component Camassa-Holm equation, viscous coupled Camassa-Holm equation, and coupled nonhomogeneous Camassa-Holm equation on periodical boundary condition.We mainly use Galerkin method, the equations we will discuss all include two variables.In the process of considering the existence of attractor for semi-group of solutions, we should take the two variables into consideration together by applying theory of Sobolev space, some PDE knowledge, Fourier restriction norm, operator, bilinear estimate, combination of energy equations and orthogonal decomposition.There are five chapters in this paper:In the first chapter, we introduce the background, status and main work.In the second chapter, we introduce the basic theory, concepts needed in the studies.In the third chapter, the Galerkin procedure is applied to show the existence of the global solution of viscous two-component Camassa-Holm equation in L2(R).The Sobolev interpolation inequality and prior estimate on time t are employed to show the existence of attracting set. Moreover, we prove that operator semi-group is compact. Finally, we get the existence of the global attractor for viscous two-component Camassa-Holm equation.In the forth, fifth chapter, we respectively focus on the existence of attractor for viscous coupled Camassa-Holm equation and coupled nonhomogeneous Camassa-Holm equation.