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

Part of: Representations of solutions Equations of mathematical physics and other areas of application Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  18 April 2023

Yanqing Wang Open the ORCID record for Yanqing Wang [Opens in a new window]  and
Gang Wu
Yanqing Wang
Affiliation:
Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, People's Republic of China wangyanqing20056@gmail.com
Gang Wu
Affiliation:
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China wugang2011@ucas.ac.cn
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Abstract

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] 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 to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ 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 $1-2s$ 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 $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–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 is under some supercritical regularity.

Keywords

Navier–Stokes equationssuitable weak solutionsbox dimensionHausdorff dimension

MSC classification

Primary: 35Q30: Navier-Stokes equations
Secondary: 35D30: Weak solutions 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 19
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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