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New Regularity Criteria Based on Pressure or Gradient of Velocity in Lorentz Spaces for the 3D Navier–Stokes Equations

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Abstract

In this paper, we derive regular criteria via pressure or gradient of velocity in Lorentz spaces to the 3D Navier–Stokes equations. It is shown that a Leray–Hopf weak solution is regular on (0, T] provided that either the norm \(\Vert \Pi \Vert _{L^{p,\infty }(0,T; L ^{q,\infty }({\mathbb {R}}^{3}))} \) with \( \frac{2}{p}+\frac{3}{q}=2\) \((\frac{3}{2}<q<\infty )\) or \(\Vert \nabla \Pi \Vert _{L^{p,\infty }(0,T; L ^{q,\infty }({\mathbb {R}}^{3}))} \) with \( \frac{2}{p}+\frac{3}{q}=3\) \((1<q<\infty )\) is small. This gives an affirmative answer to a question proposed by Suzuki (Theory Methods Appl 75:3849–3853, 2012, Remark 2.4, p. 3850). Moreover, regular conditions in terms of \(\nabla u\) obtained here generalize known ones to allow the time direction to belong to Lorentz spaces.

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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. Wang was partially supported by the National Natural Science Foundation of China under Grant (No. 11971446 and No. 11601492) and the Youth Core Teachers Foundation of Zhengzhou University of Light Industry. 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).

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Authors and Affiliations

  1. Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, Henan, People’s Republic of China

    Xiang Ji & Yanqing Wang

  2. Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi’an, 710127, Shaanxi, People’s Republic of China

    Wei Wei

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  1. Xiang Ji
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Communicated by H. Kozono.

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Ji, X., Wang, Y. & Wei, W. New Regularity Criteria Based on Pressure or Gradient of Velocity in Lorentz Spaces for the 3D Navier–Stokes Equations. J. Math. Fluid Mech. 22, 13 (2020). https://doi.org/10.1007/s00021-019-0476-8

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