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

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 $r^{-3}{\int }_{B_r^{+}}\mathrm{\mid }u(x){\mathrm{\mid }}^{3}dx$ or $r^{-2}{\int }_{B_r^{+}}\mathrm{\mid }\mathrm{\nabla }u(x){\mathrm{\mid }}^{2}dx$ 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}.