A challenging open problem: The inviscid limit under slip-type boundary conditions.

In these notes we present some results proved in the forthcoming paper [3]. We consider the $\,3-D$ evolutionary Navier-Stokes equations with a Navier slip-type boundary condition, and study the problem of the strong convergence ($ k > 1+\frac{3}{p},$ see below) of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. This problem is still open, except in the case of flat boundaries. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds. The cause of this different behavior is quite subtle.