Concerning the W k,p -Inviscid Limit for 3-D Flows Under a Slip Boundary Condition

We consider the 3-D evolutionary Navier-Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W (k, p) (Omega), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027-1055, 2007 and Beiro da Veiga and Crispo in J Math Fluid Mech, 2009, doi:10.1007/s00021-009-0295-4). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up a very elementary approach to the regularity theory, in L (p) -spaces, for solutions to the Navier-Stokes equations under slip type boundary conditions.