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

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 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 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.