\(\varepsilon \)-Regularity criteria for the 3D Navier–Stokes equations in Lorentz spaces

Abstract

In this paper, we are concerned with the regularity of suitable weak solutions to the 3D Navier–Stokes equations in Lorentz spaces. We obtain \(\varepsilon \)-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity or deformation tensor in Lorentz spaces, which generalizes the corresponding results derived by Gustafson et al. (Commun Math Phys 273:161–176, 2007) in Lebesgue spaces. As applications, this allows us to extend recent results involving Leray’s blow up rate in time, and to show that the number of singular points of weak solutions belonging to \( L^{p,\infty }(-1,0;L^{q,l}({\mathbb {R}}^{3})) \) is finite, where the pair (pq) satisfies \( {2}/{p}+{3}/{q}=1\) with \(3<q<\infty \) and \(q\le l <\infty \).

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Acknowledgements

The authors are grateful to the referee for the invaluable comments and suggestions which helped us to improve the paper significantly. The authors would like to express their sincere gratitude to Dr. Daoguo Zhou for a short discussion on Lorentz spaces. Wang was partially supported by the National Natural Science Foundation of China under grant (No. 11971446, No. 12071113 and No. 11601492). Wei was partially supported by the National Natural Science Foundation of China under grant (No. 11601423, No. 11701450, No. 11701451, No. 11771352, No. 11871057) and Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 18JK0763). Yu was partially supported by the National Natural Science Foundation of China (NNSFC) (No. 11901040), Beijing Natural Science Foundation (BNSF) (No. 1204030) and Beijing Municipal Education Commission (KM202011232020).

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Wang, Y., Wei, W. & Yu, H. \(\varepsilon \)-Regularity criteria for the 3D Navier–Stokes equations in Lorentz spaces. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00643-5

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Keywords

  • Navier–Stokes equations
  • Suitable weak solutions
  • Regularity
  • Lorentz spaces

Mathematics Subject Classification

  • 76D03
  • 76D05
  • 35B33
  • 35Q35