Partially regular weak solutions of the stationary Navier-Stokes equations in dimension 6

TL;DR

This paper establishes the existence of partially regular weak solutions to the stationary Navier-Stokes equations in six dimensions using defect measures, providing insights into their regularity and singular set size.

Contribution

It introduces a method to prove existence of partially regular solutions in 6D and estimates the singular set size using Hausdorff measures.

Findings

Existence of partially regular weak solutions in 6D with external force in specific function spaces.

Local energy estimates for these solutions.

Vanishing of defect measures under smallness conditions.

Abstract

By using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force $f \in L_{\text{loc}}^q \cap L^{3/2}, q>3$ in general open subdomains of $\mathbb{R}^6$. These weak solutions satisfy certain local energy estimates and we estimate the size of their singular sets in terms of Hausdorff measures. We also prove the defect measures vanish under a smallness condition, in contrast to the nonstationary Navier-Stokes equations in $\mathbb{R}^4 \times [0,\infty[$.