Vanishing viscosity limit for incompressible Navier-Stokes equations with Navier boundary conditions for small slip length

Abstract

In this paper, we will analyze the vanishing viscosity limit of the incompressible Navier-Stokes equations with the Navier slip boundary conditions in a bounded domain of R2. When the slip length is smaller than or equal to the order of viscosity, by using an energy method and developing Kato’s approach given in the work of Kato [Math. Sci. Res. Inst. Publ. 2, 85–98 (1984)], we obtain several conditions to guarantee that the solution of the Navier-Stokes equations with the Navier slip boundary conditions goes to the solution of the associated problem of the Euler equations in the energy space L2 uniformly in time, as the viscosity goes to zero.