Inviscid Limits For The Navier-Stokes Equations In 3D Domains With Friction Boundary Conditions

In this paper,we consider the incompressible Navier-Stokes equation with Navierfriction boundary condition in three-dimensional domains and present two results. First,inthe case of a bounded domain we prove that weak Leray solutions converge to a strongsolution of the Euler equations as the viscosity goes to 0 ,provided that the initial dataconverge to a suffciently smooth limit in L~2.In this part,we mainly apply the eigen-values and eigenfunctions of Laplace problem corresponding to the boundary conditonto apploxmate the solutions of Navier-Stokes equations. Then use series of prior esti-mate to prove the existence of weak leray solution.When viscosity goes to 0,the solutionin L~∞(0,T;L~2(Q)) and L~2(0,T;V) respectively strong and weak converge to the Eulerequations.Second,considering the case of anisotropic viscosities: fix the horizontal viscosity,letthe vertial viscosity go to 0 and prove the solutions convergence to the expected limitsystem under a weaker hypothesis on the initial data.In this part,we consider the fol-lowing three equations:anisotropic Navier-Stokes equations ,linear Navier equations andhomogeneous parabolic partical di?erence equation.Apply the basic solutions of the sec-ond equation to the original equation,then utlize the estimate of linear equations toapproxmate the inital equation.Finally we will get the results we want.