Stokes and Navier–Stokes equations with Navier boundary condition Équations de Stokes et de Navier–Stokes avec la condition de Navier

Abstract

© 2018 Académie des sciences In this paper, we study the stationary Stokes and Navier–Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Ω⊂R 3 of class C 1,1 from the viewpoint of the behavior of solutions with respect to the friction coefficient α. We first prove the existence of a unique weak solution (and strong) in W 1,p (Ω) (and W 2,p (Ω)) to the linear problem for all 1<p<∞ considering minimal regularity of α using some inf–sup condition concerning the rotational operator. Furthermore, we deduce uniform estimates of the solutions for large α 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 α→∞. Finally, we discuss the same questions for the non-linear system.