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 $Ω$ of $\mathbb{R}^n$ of class $W^{2-1/s}_s$, $s>n$. Since such domains are of class $C^{1,1-n/s}$, our result improves upon the recent one by Amrouche-Seloula, who assume $Ω$ 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 $Ω$ locally, we construct a novel $W^2_s$ diffeomorphism for $Ω$ of class $W^{2-1/s}_s$. The fractional regularity gain, from $2-1/s$ to $2$, guarantees that the Piola transform is of class $W^1_s$. This allows us to transform $W^1_r$ 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 $W^{2-1/s}_s$ seems to be minimal.