Vanishing Viscosity Limit For The Incompressible Fluid Mechanics Equations

In this paper,we mainly study the vanishing viscosity limit of the 3D nonhomo-geneous incompressible Navier-Stokes equations and the inviscid and non-resistive limit of the 3D nonhomogeneous incompressible MHD equations under certain physic boundary conditions.It is a classical issue in the mathematical study of the fluid dynamical systems,and has important physical applications and mathematical in-terests.In the presence of specific physical boundary conditions,the situation is much more complicated due to the appearance of the boundary layer.The study of viscosity vanishing limit problem has attracted much attention and the recent research has made important progress.This paper includes four main parts to explain some basic properties and basic theory of incompressible fluid dynamic equations as follows:In Chapter 2,we discuss the vanishing viscous limit problem for incompressible Navier-Stokes equations in the recent progress;Hodge decomposition theory;Laplace operator,Stokes operator and compact operator theory and some main convergence results related with our work.In Chapter 3,we introduce the progress of the investigation on the vanishing viscosity limit of the nonhomogeneous incompressible Navier-Stokes equations in the case of the whole space,bounded region with No-Slip boundary conditions or general Navier-Slip boundary conditions.In Chapter 4,we consider the viscosity vanishing limit of nonhomogeneous incompressible Navier-Stokes equations with boundary conditions,three differen-t cases as follows:Firstly,the case where the region is flat domain is considered,on the premise that the norm direction of the initial gradient density vanishes,we obtained a uniform bounded estimate of the 3D nonhomogeneous incompressible Navier-Stokes equations and the strong convergence of the solutions in the sense of H~2(?)norm;Then,we consider the case where the region is a general smooth bound-ed region with curvature,we obtained the strong convergence of the solutions in the sense of H~1(?)norm based on the initial gradient density and the initial vorticity vanishing on the boundary.Last,as the boundary condition is the Vortisity-Slip boundary condition,under the condition that the initial gradient density vanishes,we obtained the strong convergence results in the sense of H~1(?)norm for the 3D nonhomogeneous incompressible Navier-Stokes equations.In Chapter 5,we consider the vanishing viscosity limit of the 3D nonhomoge-neous incompressible MHD equations with Slip boundary conditions.We proved that the region must be flat domain and obtained the strong convergence results;However,in the case of a general smooth bounded region with curvature,The strong convergence results can not be obtained as nonhomogeneous incompressible Navier-Stokes equations due to the momentum equations are coupled with the momentum and the magnetic field.In Chapter 6,mainly summarizes the research results and significance.At the same time,it puts forward some restrictions of the technical in the research process.The vanishing viscosity limit problem under physical boundary conditions is expected to be solved fundamentally by different mathematical tools.