Boundary partial regularity for the high dimensional Navier-Stokes equations

TL;DR

This paper proves that the set of singular points has zero two-dimensional Hausdorff measure at the boundary for suitable weak solutions of high-dimensional Navier-Stokes equations, in both 4D time-dependent and 6D stationary cases.

Contribution

It establishes boundary partial regularity results for high-dimensional Navier-Stokes equations, extending regularity theory to 4D and 6D cases.

Findings

Zero two-dimensional Hausdorff measure of singular points at the boundary

Boundary regularity results for 4D time-dependent Navier-Stokes

Boundary regularity results for 6D stationary Navier-Stokes

Abstract

We consider suitable weak solutions of the incompressible Navier--Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero in both cases.