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. In this paper, we study the vanishing viscosity limit for the incompressible NavierStokes equations with the Navier friction boundary condition. To simplify the expansion of solutions in terms of the viscosity, we shall only consider the case that the slip length α in the Navier boundary condition is a power of the viscosity ǫ, α= ǫ . First, by multi-scale analysis we formally deduce that γ= 1 2 is critical in determining the boundary layer behavior. When γ > 1 2 , the boundary layer appears in the zero-th order terms of the expansion of solutions, and satisfies the same boundary value problem for the nonlinear Prandtl equations as in the non-slip case, when γ= 1 2 , the boundary layer also appears in the zero-th order terms of solutions, and satisfies the nonlinear Prandtl equations but with a Robin boundary condition for the tangential velocity profile, and when γ < 1 2 , the boundary layer appears in the order O(ǫ) terms of solutions, and satisfies a boundary value problem for the linearized Prandtl equations. Secondly, we justify rigorously the asymptotic behavior of the vanishing viscosity limit for the incompressible Navier-Stokes equations with anisotropic viscosities by using the energy method, when the slip length is larger than the square root of the vertical viscosity. Even though the boundary layer appears in the lower order terms of solutions and satisfies a linear problem, the vorticity of flow is unbounded in the vanishing viscosity limit.

Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit