Fractal dimension of potential singular points set in the Navier-Stokes equations under supercritical regularity

TL;DR

This paper investigates the fractal dimensions of potential singular points in solutions to the Navier-Stokes equations under supercritical regularity, providing bounds on their box and Hausdorff dimensions.

Contribution

It establishes new upper bounds on the fractal dimensions of singular sets for Navier-Stokes solutions with supercritical regularity, answering open questions in the field.

Findings

Upper box dimension of singular points set is at most a specific function of p and q.

Hausdorff measure of singular points set is zero under certain regularity conditions.

Results extend understanding of singular set structure in supercritical regimes.

Abstract

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [21, p. 505, Comm. Math. Phys., 2010]{[RS3]} for the Navier-Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal{S}$ of suitable weak solution $u$ belonging in $ L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for $1\leq\frac{2}{q}+\frac{ 3}{p}\leq\frac32$ with $2\leq q<\infty$ and $2<p<\infty$ is at most $\max\{p,q\}(\frac{2}{q}+\frac{ 3}{p}-1)$ in this system. Secondly, it is shown that $1-2 s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $ u\in L^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ for $0\leq s\leq\frac12$ is zero, whose proof relies on Caffarelli-Silvestre's extension. Inspired by Baker-Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity under some supercritical regularity.