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 H\"{o}lder continuous at $0$ provided that $\int_{B_1}|u(x)|^3dx+\int_{B_1}|f(x)|^qdx$ or $\int_{B_1}|\nabla u(x)|^2dx$+$\int_{B_1}|\nabla u(x)|^2dx\left(\int_{B_1}|u(x)|dx\right)^2+\int_{B_1}|f(x)|^qdx$ 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 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.