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

TL;DR

This paper establishes well-posedness for the stationary Stokes problem with Navier slip boundary conditions on domains with minimal fractional Sobolev regularity, using novel localization and boundary flattening techniques.

Contribution

It introduces a new approach to handle Navier slip boundary conditions on less regular domains, improving previous regularity assumptions from $C^{1,1}$ to fractional Sobolev spaces.

Findings

Proves well-posedness in reflexive Sobolev spaces for domains of class $W^{2-1/s}_s$.

Develops a new $W^2_s$ diffeomorphism for boundary flattening.

Ensures the Piola transform preserves regularity and boundary normal vector.

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 $W^{2-1/s}_s$, $s>n$. Since such domains are of class $C^{1,1-n/s}$, our result improves upon the recent one by Amrouche-Seloula, who assume $\Omega$ 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 $W^2_s$ diffeomorphism for $\Omega$ 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 H\"older. It is in this sense that the boundary regularity $W^{2-1/s}_s$ seems to be minimal.