Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions

We consider the Navier-Stokes equations with Navier friction boundary conditions and prove two results. First, in the case of a bounded domain we prove that weak Leray solutions converge (locally in time in dimension ≥ 3 and globally in time in dimension 2) as the viscosity goes to 0 to a strong solution of the Euler equations provided that the initial data converges in L2 to a sufficiently smooth limit. Second, we consider the case of a half-space and anisotropic viscosities: we fix the horizontal viscosity, we send the vertical viscosity to 0 and prove convergence to the expected limit system under weaker hypothesis on the initial data. Introduction We consider in this paper the vanishing viscosity limit for the incompressible NavierStokes equations in a domain Ω: ∂tu− ν4u+ u · ∇u = −∇p, in Ω× (0,+∞), div u = 0, in Ω× [0,+∞), u ∣∣ t=0 = u0, in Ω. (1) The vanishing viscosity limit for the incompressible Navier-Stokes equations, in the case where there exist physical boundaries, is a challenging problem due to the formation of a boundary layer which is caused by the classical no-slip boundary condition. A partial result, in the case of half-space, was given in [17, 18] by imposing analyticity on the initial data. The authors proved in these papers that the Navier-Stokes solution goes to an Euler solution outside a boundary layer, and it is close to a solution of the Prandtl equations within the boundary layer. Concerning the anisotropic Navier-Stokes equations, in some particular domains such as the half-space, it was showed in [10] that if the ratio of vertical viscosity to horizontal viscosity also goes to zero, then the weak solutions converge to the solution of the Euler system. ∗Partially supported by FAPESP, grant 02/13137-0