Interior and Boundary Regularity Criteria for the 6D steady Navier-Stokes Equations

TL;DR

This paper establishes interior and boundary regularity criteria for suitable weak solutions to the 6D steady Navier-Stokes equations, showing regularity under smallness conditions on certain integral norms, thus improving previous results.

Contribution

It introduces new smallness conditions involving integral norms that ensure regularity of solutions, extending prior regularity criteria for the 6D steady Navier-Stokes equations.

Findings

Solutions are Hölder continuous at zero under small integral norms.

The set of singular points has zero 2D Hausdorff measure.

Regularity criteria are improved over previous theorems.

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.