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On Wolf’s Regularity Criterion of Suitable Weak Solutions to the Navier–Stokes Equations

  • Quansen Jiu
  • Yanqing WangEmail author
  • Daoguo Zhou
Article

Abstract

In this paper, we consider the local regularity of suitable weak solutions to the 3D incompressible Navier–Stokes equations. By means of the local pressure projection introduced by Wolf (in: Rannacher, Sequeira (eds) Advances in mathematical fluid mechanics, Springer, Berlin, 2010, Ann Univ Ferrara 61:149–171, 2015), we establish a Caccioppoli type inequality just in terms of velocity field for suitable weak solutions to this system
$$\begin{aligned} \Vert u\Vert ^{2}_{L^{\frac{20}{7},\frac{15}{4}}Q(\frac{1}{2})}+ \Vert \nabla u\Vert ^{2}_{L^{2}(Q(\frac{1}{2}))} \le C \Vert u\Vert ^{2}_{L^{\frac{20}{7}}(Q(1))}+C\Vert u\Vert ^{4}_{L^{\frac{20}{7}}(Q(1))}. \end{aligned}$$
This allows us to derive a new \(\varepsilon \)-regularity criterion: Let u be a suitable weak solution in the Navier–Stokes equations. There exists an absolute positive constant \(\varepsilon \) such that if u satisfies
$$\begin{aligned} \iint _{Q(1)}|u|^{20/7}dxdt< \varepsilon , \end{aligned}$$
then u is bounded in some neighborhood of point (0, 0). This gives an improvement of previous corresponding results obtained in Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), in Guevara and Phuc (Calc Var 56:68, 2017) and Wolf (Ann Univ Ferrara 61:149–171, 2015).

Keywords

Navier–Stokes equations Suitable weak solutions Regularity 

Mathematics Subject Classification

76D03 76D05 35B33 35Q35 

Notes

Acknowledgements

We are deeply grateful to the anonymous referee and the associated editor for the invaluable comments and suggestions which helped to improve the paper significantly. Jiu was partially supported by the National Natural Science Foundation of China (No. 11671273) and by Beijing Natural Science Foundation (No. 1192001). 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). Part of this work was done when D. Zhou is visiting Mathematical Institute of University of Oxford. Zhou appreciates Prof. G. Seregin’s warm hospitality and support.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

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Authors and Affiliations

  1. 1.School of Mathematical SciencesCapital Normal UniversityBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and Information ScienceZhengzhou University of Light IndustryZhengzhouPeople’s Republic of China
  3. 3.School of Mathematics and Information ScienceHenan Polytechnic UniversityJiaozuoPeople’s Republic of China

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