Study On Limits For Several Fluid Equations

In this paper,we consider several common fluid equations,whose strong solutions and related limit problems are studied,that is to say,Our study is about the vanishing viscosity limit of local solution and decay of global solution.To be more precise,the vanishing viscosity limit problem is that when the viscous coefficient or diffusion coefficient approaches zero,the solution of viscous fluid equations converge to the solutions of inviscid or ideal fluid equations.In bounded domain,the boundary conditions will be a key,here we consider slip boundary condition.Decay of global solutions means when time tends to infinity,the energy tends to zero,which is a large-time behavior,including decay rate and asymptotic stability problems.We consider the decay in L2 for weak or strong solution of fluid equations in R2 or R3 by the conventional Fourier separation method.Recently,Many authors are interested in two kinds of problems to hydrodynamic equations,we also have carried out a series of studies about these.In chapter 1,we give a brief description of the background of several fluid models.Next,we recall some results which related issues on our research,and give a detailed description of the current situation of the development.Finally,the arrangement of this paper is briefly explained.In Chapter 2,we consider vanishing viscosity limit problem of three-dimensional incompressible Navier-Stokes equations under the slip boundary condition.In bounded domain with flat boundary,we improve the regularity of strong solution by Lp theory and previous results.Then the uniform bounds are obtained by substitution,and thus the convergence of strong solutions is established.In Chapter 3,we study the initial-boundary value problem for three-dimensional nonhomogeneous incompressible MHD equations with slip boundary conditions in bounded domain.It is worth mentioning that we only need general regularity of the initial value without compatibility condition.The existence and uniqueness of local strong solutions are obtained by time weighted estimations.Finally,we discuss the vanishing viscosity limit problem.Because there is no uniform bound for strong solutions,therefore,we require that the strong solutions of ideal MHD systems possess higher regularity.In Chapter 4,we are also interested in the study of three-dimensional incom-pressible Boussinesq equations.Firstly,we study local existence of strong solution to the equations in general bounded domain under slip boundary condition.Next,we obtain the uniform bounds of strong solutions by energy method,The vanishing viscosity limit problem is considered,and it can be proved by two aspects:vanishing partial viscosity and vanishing complete viscosity.In Chapter 5,we focus on the decay of strong solutions of two-dimensional trop-ical climate model.We start with the establishment of weak solution decay,and obtain the decay of the strong solution later,finally,we extend this conclusion to any order.In Chapter 6,we devote to the global regularity of solutions of generalized mi-cropolar equations.The global strong solution of three-dimensional micropolar equa-tions is open as the three-dimensional Navier-Stokes equations.In general Sobolev space,we improve the order of diffusion term,it follows that global strong solution is obtained and the existence of classical solution is also proved.In Chapter 7,we will discuss the large-time behavior of the micropolar equations with nonlinear damping term.Semigroup method cannot be used directly because of some linear terms.We via Fourier transform to overcome the difficulty.The decay rates of weak solutions and strong solutions are proved by Fourier splitting method.The asymptotic stability is also discussed with initial perturbations,and the error between the solution of classical micropolar equations is established.