On the box-counting dimension of potential singular set for suitable weak solutions to the 3D Navier-Stokes equations

TL;DR

This paper improves the upper bound on the box-counting dimension of potential singular points in 3D Navier-Stokes solutions from approximately 1.55 to 1.30 by leveraging pressure estimates.

Contribution

It introduces a novel approach using pressure in terms of its gradient to tighten the upper bound on the singular set dimension.

Findings

Upper box-counting dimension is at most 135/104 (~1.30).

Improves previous bounds of 95/63 (~1.51) and 45/29 (~1.55).

Advances understanding of singularity structure in Navier-Stokes solutions.

Abstract

In this paper, we are concerned with the upper box-counting dimension of the set of possible singular points in space-time of suitable weak solutions to the 3D Navier-Stokes equations. By taking full advantage of the pressure $\Pi$ in terms of $\nabla \Pi$ in equations, we show that this upper box dimension is at most $135/104(\approx1.30)$, which improves the known upper box-counting dimension $95/63(\approx1.51)$ in Koh et al. [9, J. Differential Equations, 261: 3137--3148, 2016], $45/29(\approx1.55)$ in Kukavica et al. [11, Nonlinearity 25: 2775-2783, 2012] and $135/82(\approx1.65)$ in Kukavica [10, Nonlinearity 22: 2889-2900, 2009].