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 ${\int }_{B_1}|u(x){|}^{3}dx+{\int }_{B_1}|f(x){|}^{q}dx$ or ${\int }_{B_1}|\nabla u(x){|}^{2}dx$+${\int }_{B_1}|\nabla u(x){|}^{2}dx{\left({\int }_{B_1}|u(x)|dx\right)}^2+{\int }_{B_1}|f(x){|}^{q}dx$ 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 obtain that $0$ is regular provided that ${\int }_{B_1^{+}}|u(x){|}^{3}dx+{\int }_{B_1^{+}}|f(x){|}^{3}dx$ or ${\int }_{B_1^{+}}|\nabla u(x){|}^{2}dx+{\int }_{B_1^{+}}|f(x){|}^{3}dx$ is sufficiently small. These results improve previous regularity theorems by Dong-Strain (\cite{DS}, Indiana Univ. Math. J., 2012), Dong-Gu (\cite{DG2}, J. Funct. Anal., 2014), and Liu-Wang (\cite{LW}, J. Differential Equations, 2018), where either the smallness of the pressure or the smallness on all balls is necessary.

arXiv commentarXiv admin note: text overlap with arXiv:1309.3158