Boundary Regularity Criteria for the 6D Steady Navier-Stokes and MHD Equations

TL;DR

This paper establishes boundary regularity criteria for 6D steady Navier-Stokes and MHD equations, showing solutions are Hölder continuous near boundaries under certain smallness conditions, thus extending interior regularity results.

Contribution

It introduces new boundary regularity criteria for high-dimensional steady Navier-Stokes and MHD equations, generalizing previous interior regularity findings.

Findings

Solutions are Hölder continuous near boundary under smallness conditions.

The set of boundary singular points has zero 2D Hausdorff measure.

Extends interior regularity results to boundary cases.

Abstract

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes and MHD equations are H\"older continuous near boundary provided that either $r^{-3}\int_{B_r^+}|u(x)|^3dx$ or $r^{-2}\int_{B_r^+}|\nabla u(x)|^2dx$ is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by Dong-Strain \cite{DS}.