The Stokes problem with Navier slip boundary condition: Minimal fractional Sobolev regularity of the domain

Abstract

We prove well-posedness in reflexive Sobolev spaces of weak solutions to the stationary Stokes problem with Navier slip boundary condition over bounded domains $\Omega$ of ${\mathbb{R}}^n$ of class $W_s^{2-1/s}$, $s>n$. Since such domains are of class $C^{1,1-n/s}$, our result improves upon the recent one by Amrouche-Seloula, who assume $\Omega$ to be of class $C^{1,1}$. We deal with the slip boundary condition directly via a new localization technique, which features domain, space and operator decompositions. To flatten the boundary of $\Omega$ locally, we construct a novel $W_s^2$ diffeomorphism for $\Omega$ of class $W_s^{2-1/s}$. The fractional regularity gain, from $2-1/s$ to $2$, guarantees that the Piola transform is of class $W_s^1$. This allows us to transform $W_r^1$ vector fields without changing their regularity, provided $r\le s$, and preserve the unit normal which is Hölder. It is in this sense that the boundary regularity $W_s^{2-1/s}$ seems to be minimal.