# Improved bounds for box dimensions of potential singular points to the Navier--Stokes equations

## TL;DR

This paper establishes improved upper bounds on the box dimensions of potential singular points in solutions to the 3D Navier--Stokes equations, advancing understanding of their regularity properties.

## Contribution

It provides new bounds on the box dimensions of interior and boundary singular points using recent $ ext{epsilon}$-regularity criteria, improving previous estimates.

## Key findings

- Interior singular points have box dimension at most 7/6.
- Boundary singular points have box dimension at most 10/9.
- Proofs leverage recent progress in $ ext{epsilon}$-regularity at one scale.

## Abstract

In this paper, we study the potential singular points of interior and boundary suitable weak solutions to the 3D Navier--Stokes equations. It is shown that upper box dimension of interior singular points and boundary singular points are bounded by $7/6$ and $10/9$, respectively. Both proofs rely on recent progress of $\varepsilon$-regularity criteria at one scale.