The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer

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. The crucial point here is that the boundary is not flat. In fact (see Beirão da Veiga et al. in J Math Anal Appl 377:216–227, 2011) if $${\,\Omega = \mathbb R^3_+,}$$ convergence holds in $${C([0,T]; W^{k,p}(\mathbb R^3_+))}$$ , for arbitrarily large k and p. For this reason, the negative answer given here was not expected.