Stokes and Navier-Stokes equations with Navier boundary condition

In this paper, we study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Omega subset of R-3 of class C-1,C-1 from the viewpoint of the behavior of solutions with respect to the friction coefficient alpha. We first prove the existence of a unique weak solution (and strong) in W-1,W-P(Omega) (and W-2,W-P(Omega)) to the linear problem for all 1 < p < infinity considering minimal regularity of alpha, using some inf-sup condition concerning the rotational operator. Furthermore, we deduce uniform estimates of the solutions for large alpha, which enables us to obtain the strong convergence of Stokes solutions with Navier slip boundary condition to the one with no-slip boundary condition as alpha -> infinity. Finally, we discuss the same questions for the non-linear system. (C) 2018 Academie des sciences. Published by Elsevier Masson SAS.