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

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Holder continuous at 0 provided that fB1 |u(x)|3dx+fB1 |f (x)|qdx or fB1 | backward difference u(x)|2dx+fB1 | backward difference u(x)|2dx x (f )2 + f B1 |u(x)|dxB1 |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 pro-vided that fB1+ |u(x)|3dx+fB1+ |f(x)|3dx or fB1+ | backward difference u(x)|2dx +fB1+ |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. (c) 2022 Elsevier Inc. All rights reserved.