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We show that, in general, the solutions to the initial-boundary value problem for the Navier-Stokes equations under a widely adopted Navier-type slip boundary condition do not converge, as the viscosity goes to zero, to the solution of the Euler equations under the classical zero-flux boundary condition, and same smooth initial data, in any arbitrarily small neighborhood of the initial time. Convergence does not hold with respect to any space-topology which is sufficiently strong as to imply that the solution to the Euler equations inherits the complete slip type boundary condition. In our counter-example Ω is a sphere, and the initial data may be infinitely differentiable.

We consider a stationary Navier-Stokes system with shear dependent viscosity, under Dirichlet boundary conditions. We prove Hölder continuity, up to the boundary, for the gradient of the velocity field together with the L 2 -summability of the weak second derivatives. The results hold under suitable smallness assumptions on the force term and without any restriction on the range of p ∈ (1, 2).

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