Global Attractor And Exponent Attractor Of2n-Order Boussinesq Equation

In this paper, firstly, we using the Galerkin method combined with Growall inequality to study a2n-order Boussinesq equation with Dirichlet boundary conditions. Existence, uniqueness and continuous dependence upon data of a global solution of the system in certain initial conditions and Dirichlet boundary conditions are proved. Secondly, followed by a prior estimate combined with some inequalities skills and related theory of Sobolev space, the existence of bounded absorbing sets is proved; Finally, using the existence of global attractor and exponential attractor theorem, prove the existence of global attractor and exponential attractor of the system. Specific content as follows:In the first chapter, we introduce the background, development and status of the Boussi-nesq equation, and we give the main results of this thesis.In the second chapter, basic knowledge, mainly introduce the basic concepts, inequality and lemma needed in the studies.In the third chapter, study the initial boundary value problem of the2n-order Boussinesq equation with the initial conditions and the boundary conditions or in which that Ω is a bounded set with smooth boundary in RN and△is the Laplacian. by using the Galerkin theory,we get the existence, uniqueness and continuous dependence upon data of a global solution for the system.In the forth chapter, study the initial boundary value problem of the2n-order Boussinesq Equation with the initial conditions and the boundary conditions or The existence of the global attractor of this system are proved. In this paper we using prior estimate on time t are employed to show the existence of attracting set for the solution semigroups in the space E1under certain conditions. Moreover, we prove that operator semi-group is compact. Finally, we get the existence of the global attractor for the system we considered.In the fifth chapter, study the existence of exponential attractor of the2n-order Boussi-nesq equation under certain initial conditions and the two Dirichlet boundary conditions.