Interior and boundary regularity criteria for the 6D steady Navier-Stokes equations

Abstract

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫B1|u(x)|3dx+∫B1|f(x)|qdx or ∫B1|∇u(x)|2dx+∫B1|∇u(x)|2dx(∫B1|u(x)|dx)2+∫B1|f(x)|qdx with q>3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫B1+|u(x)|3dx+∫B1+|f(x)|3dx or ∫B1+|∇u(x)|2dx+∫B1+|f(x)|3dx is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8], Indiana Univ. Math. J., 2012), Dong-Gu ([7], J. Funct. Anal., 2014), and Liu-Wang ([27], J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary.