The Stokes problem with Navier boundary conditions in irregular domains

TL;DR

This paper establishes maximal regularity estimates for the steady Stokes equations with Navier boundary conditions in irregular domains, highlighting the minimal regularity needed and demonstrating the sharpness of these results.

Contribution

It provides the first maximal regularity results for the Stokes problem with Navier boundary conditions in domains with minimal regularity, including sharpness examples.

Findings

Maximal regularity estimates in $W^{1,p}$ and $W^{2,p}$ spaces for Navier boundary conditions.

One derivative more is needed for boundary charts compared to no-slip conditions.

Results are sharp, as demonstrated by specific examples.

Abstract

We consider the steady Stokes equations supplemented with Navier boundary conditions including a non-negative friction coefficient. We prove maximal regularity estimates (including the prominent spaces $W^{1,p}$ and $W^{2,p}$ for $1<p<\infty$ for the velocity field) in bounded domains of minimal regularity. Interestingly, exactly one derivative more is required for the local boundary charts compared to the case of no-slip boundary conditions. We demonstrate the sharpness of our results by a propos examples.