Global Well-posedness Of Solutions Of Boussinesq Equation With Partial Dissipation

Hydrodynamics is one of the important contents of the study of PDEs.The main problems come from geophysics,electromagnetics,ocean fluid and atmospheric science Boussinesq equations,which are physical models to describe the atmosphere encounter turbulence,since Alfven first discussed the linear stability of the background magnetic field when there is no magnetic diffusion,the stability and large time behavior of the solution after the disturbance near the equilibrium state have attracted the attention of many mathematicians.This paper introduces the attention and research status of the well posedness Boussinesq equation with partial dissipation.This paper also introduces the study of the stability and large time behavior of the Boussinesq equation with partial dissipation near the equilibrium state.It also discusses the existence and uniqueness of the weak solutions of the Boussinesq equation system with fractional dissipation as follows:In Chapter 1,we introduce the signification of the global well posedness of the solution for Boussinesq equation with partial dissipation and expound the research status of different types of Boussinesq equation.Furthermore,we discuss the main research contents of this paper and state the marks of this paper.In Chapter 2,we summarize the basic theoretical knowledge,including inequality estimation,functional space,Fourier transform and Boussinesq equation classifcation.In Chapter 3,we discuss the perturbed Boussinesq equation with partial dissipation near the equilibrium state.By using anisotropic estimation,bootstrap principle and qualitative theory in ordinary differential equations,we prove the stability of the solution in H1 and H2 space respectively.The uniqueness of the solution is proved by the norm of the difference between any two solutions.In Chapter 4,the linear part of the perturbed system is first obtained and the decay rate of the linear part is calculated by plancherel's inequality and energy method.In Chapter 5,we will establish the local existence and uniqueness when the initial data(?)for the largest possible range of a and ?.The uniqueness of weak solution can be directly established by the norm of the difference between any two weak solutionsAt last,we give a summarize and put forward some prospects for the future works.