Partial boundary regularity for the Navier-Stokes equations in irregular domains

TL;DR

This paper establishes partial boundary regularity for solutions to the Navier-Stokes equations in irregular domains with minimal regularity assumptions, extending previous results from smoother to less regular boundaries.

Contribution

It introduces a new criterion for boundary regularity and proves existence of solutions continuous at almost all boundary points under weak boundary regularity conditions.

Findings

Boundary regularity criterion for Navier-Stokes solutions

Existence of solutions continuous at almost all boundary points

Extension of regularity results to domains with minimal boundary regularity

Abstract

We prove partial regularity of suitable weak solutions to the Navier--Stokes equations at the boundary in irregular domains. In particular, we provide a criterion which yields continuity of the velocity field in a boundary point and obtain solutions which are continuous in a.a. boundary boundary point (their existence is a consequence of a new maximal regularity result for the Stokes equations in domains with minimal regularity). We suppose that we have a Lipschitz boundary with locally small Lipschitz constant which belongs to the fractional Sobolev space $W^{2-1/p,p}$ for some $p>\frac{15}{4}$. The same result was previously only known under the much stronger assumption of a $C^2$-boundary.