Vanishing Viscosity Limit For Two Classes Of Fluid Mechanics Equations

In this thesis,we study the regularity and the vanishing viscosity limit of solutions for the three dimensional incompressible Boussinesq equations and three dimensional inhomogeneous incompressible MHD-Boussinesq equations under Slip boundary conditions in a bounded domain.When the diffusivity or viscosity coefficient approaches to zero,the solutions of the equations converge to the solutions of an ideal equations without diffusivity or viscosity.In Chapter 3,we study the vanishing diffusivity or viscosity limit of three dimensional incompressible Boussinesq equations with Slip boundary conditions in Lp(p>3)spaces when the bounded domain is flat.We obtain the higher order energy inequalities of the equations by using the energy estimates.By constructing a new ordinary differential equation,we overcome the difficulties brought by the nonlinear terms,and get the local bounds of ‖u‖W3,p and ‖θ‖W3,p.We further consider the convergence rates of the velocity field and temperature in three cases that the thermal diffusivity approaches to zero,the viscosity approaches to zero,and both of them approach to zero.In Chapter 4,we study the vanishing diffusivity limit of three dimensional incompressible Boussinesq equations with Slip boundary conditions in Lp(p>3)spaces in the general smooth bounded domain.We use the energy estimates to obtain that the solutions of the Boussinesq equations converge to the solutions of the ideal Boussinesq equations when the thermal diffusivity approaches zero.At the same time,we get the convergence rates of the velocity field and temperature.In Chapter 5,we study the vanishing diffusivity limit of inhomogeneous MHD-Boussinesq equations in L2 space.When the velocity field and magnetic field satisfy the Slip boundary conditions,and the temperature satisfies the Neumann boundary condition,we overcome the difficulties of the boundary terms and the nonlinear terms,and establish the local regularity of the solutions by energy estimates.In addition,the convergence rates of density,velocity field,magnetic field and temperature are obtained as the thermal diffusivity approaches to zero.