Well-posedness For Several Classes Of Equations In Fluid Mechanics

The main results included here are concerned with the well-posedness of any given solution to several classes of equations in fluid mechanics.The layout of this thesis is as follows.We introduce in Chapter 1 the background and research progresses about the incompressible Boussinesq equations,and recall some preliminaries which include some notations,useful inequalities and function spaces(particularly,Besov spaces are based on the Littlewood-Paley theory).Chapter 2 deals with the well-posedness of the 1D transport equation with nonlocal velocity in the Lei-Lin space.We establish the nonlinear estimates by means of splitting frequency technique,which is crucial to obtain the a priori estimates.By the standard compactness argument,we prove that the 1D transport equation has a unique global solution with small initial data and the solution is stable.Chapter 3 is devoted to the study of the 2D incompressible liquid crystal equations.Firstly,we obtain the existence and uniqueness of global smooth solutions to the 2D incompressible liquid crystal equations with weak velocity dissipation with the aid of nonlinear maximum principle and the standard energy method;Secondly,we prove that there exists a unique global solution with small initial data for the Cauchy problem of the 2D incompressible liquid crystal equations with damping effect.Chapter 4 is concerned with the Cauchy problem for the 2D incompressible Boussinesq equations with variable viscosity and damping effect.The obvious advantage of the damping term lies in that it provides exponential decay of||?||_Lp,which has been sufficiently explored.However,the main difficulty to handle the Navier-Stokes system with variable viscosity(depending on?)lies in the propagation of the regularity of?.Taking advantage of the micro-localization and splitting frequency technique,we present the global well-posedness of the Cauchy problem.Finally,in the last chapter,we consider the initial boundary value problem for the 3D Boussinesq system with the thermal damping.By the Schauder fixed point theorem,we first establish the global existence of weak solutions,then obtain the higher regularity for the weak solutions with some small initial data,where the crucial ingredient is to utilize the polynomial decay-in-time estimate for small time and the exponential decay-in-time estimate for large time with respect to the velocity field.