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Journal of Nonlinear Science

, Volume 29, Issue 6, pp 2681–2698 | Cite as

New \(\varepsilon \)-Regularity Criteria of Suitable Weak Solutions of the 3D Navier–Stokes Equations at One Scale

  • Cheng He
  • Yanqing WangEmail author
  • Daoguo Zhou
Article

Abstract

In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new \(\varepsilon \)-regularity criteria for suitable weak solutions of the 3D Navier–Stokes equations at one scale: for any \(p,q\in [1,\infty ]\) satisfying \(1\le 2/q+3/p <2\), there exists an absolute positive constant \(\varepsilon \) such that \(u\in L^{\infty }(Q(1/2))\) if
$$\begin{aligned} \Vert u\Vert _{L^{p,q}(Q(1))}+\Vert \Pi \Vert _{L^{1 }(Q(1))}<\varepsilon . \end{aligned}$$
This is an improvement of corresponding results recently proved by Guevara and Phuc (Calc Var 56:68, 2017). As an application of these \(\varepsilon \)-regularity criteria, we improve the known upper box dimension of the possible interior singular set of suitable weak solutions of the Navier–Stokes system from \(975/758(\approx 1.286)\) (Ren et al. in J Math Anal Appl 467:807–824, 2018) to \(2400/1903 (\approx 1.261)\).

Keywords

Navier–Stokes equations Suitable weak solutions Regularity Box dimension 

Mathematics Subject Classification

35B65 35D30 76D05 

Notes

Acknowledgements

The research of Wang was partially supported by the National Natural Science Foundation of China under Grant No. 11601492 and the Youth Core Teachers Foundation of Zhengzhou University of Light Industry. The research of Zhou is supported in part by the China Scholarship Council for one-year study at Mathematical Institute of University of Oxford and Doctor Fund of Henan Polytechnic University (No. B2012-110).

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Copyright information

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Authors and Affiliations

  1. 1.Division of Mathematics, Department of Mathematical and Physical SciencesNational Natural Science Foundation of ChinaBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and Information ScienceZhengzhou University of Light IndustryZhengzhouPeople’s Republic of China
  3. 3.College of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoPeople’s Republic of China

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