Existence And Decay Estimates Of Solutions For The 3D Incompressible Boussinesq-MHD System

In this thesis,the existence and decay estimates of solutions for three dimen-sional incompressible Boussinesq-MHD system are studied.The main contents of this thesis are as follows.The first content is to investigate the existence of solutions for three-dimensional incom-pressible Boussinesq-MHD system with Navier-slip boundary conditions in a smooth bounded region.Firstly,we use Galerkin approximation to construct the sequence of approximate solu-tion of the Boussinesq-MHD system.Then,the global existence of the weak solution is proved by the uniform boundedness of the approximate solution sequences and the weak compact-ness of Sobolev space in the sense of H~1norm.Finally,the local existence and uniqueness of the strong solution of the system are proved by combining with the standard limit process,Gronwall inequality and initial conditions.The second content is to investigate the decay rates of the solutions for three-dimensional incompressible Boussinesq-MHD system in full space.Firstly,the energy inequality of the higher-order spatial derivatives of the solutions are obtained by a priori estimation.Then,com-bining the estimation of the nonlinear term in the negative Sobolev space,we obtain the energy estimates of the negative Sobolev norm of the solutions.Finally,the decay rates of the each derivative of the solutions are obtained through the Gronwall inequality.