On The Vanishing Viscosity Limit For The Incompressible Navier-Stokes Equations With The Navier Boundary Conditions

The study on the small viscosity limit of the incompressible Navier-Stokes equations is a very important research topic,both in the practical application and in the theory of the partial differential equations.To study the small viscosity limit for the flows in a region with physical boundaries not only needs to consider the behavior of the boundary layers,but also needs to analyse when the solutions of the Euler equations approximately describe the solutions of the Navier-Stokes equations.There are many interesting works for the problem with the non-slip boundary conditions.However,there is little study on the problems with the Navier slip boundary conditions,especially for the case that the slip coefficient depends on viscosities.Usually in physical models,the slip length not only depends on the curvature of physical boundary,but also is related to the viscosity coefficient of the fluid.Therefore,the study on the small viscosity limit problem with the Navier slip boundary conditions is very challenging and interesting.In this thesis,we will analyze the small viscosity limit problem of the incompressible Navier-Stokes equations with the Navier slip boundary conditions.For the different kinds of the dependence of the slip coefficient on viscosities,we will establish the various con-ditions to guarantee that the solutions of the Navier-Stokes equations can be approximately described by the solutions of the corresponding Euler equations in the small viscosity limit.We first give the definition of the weak solutions of the incompressible Navier-Stokes equa-tions with the Navier boundary conditions and the smooth solutions of the corresponding Euler equations.Then we give a relative energy inequality for the weak solutions of this Navier-Stokes equations.Then through the development of Kato's method,by constructing an artificial boundary layer,and using the analysis tool of the Young inequality,Holder in-equality and Poincare inequality etc.,we get the conclusions that for the case that the slip coefficient is suitably large,the solutions of the Navier-Stokes equations with the Navier slip boundary conditions are approximated by the solutions of the associated problem of the Eu-ler equations without any extra conditions,for the case that the slip coefficient is relatively small,we get four sufficient conditions.These conditions characterize the energy dissipation in the boundary layer.Under one of these four conditions,it is concluded that the solution-s of the Navier-Stokes equations with the Navier slip conditions are approximated by the solutions of the corresponding Euler equations in the energy space.