On regularity criteria for the Navier–Stokes equations based on one directional derivative of the velocity or one diagonal entry of the velocity gradient

Abstract

It is proved that if the solution of the Navier–Stokes system satisfies

$$\begin{aligned} \partial _3\varvec{u}\in L^p(0,T;L^q(\mathbb {R}^3)),\quad \frac{2}{p}+\frac{3}{q} =\frac{22}{13}+\frac{3}{13q},\quad 3<q<4, \end{aligned}$$

or

$$\begin{aligned} \partial _3u_3\in L^\beta (0,T;L^\alpha (\mathbb {R}^3)),\quad \frac{2}{\beta }+\frac{3}{\alpha } =\frac{3(\sqrt{65\alpha ^2-78\alpha +49}+7-\alpha )}{16\alpha },\quad \frac{3+\sqrt{17}}{4}\le \alpha \le \infty , \end{aligned}$$

then the solution is smooth on (0, T]. These two improve many previous results.

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (Grant No. 11761009) and the Natural Science Foundation of Jiangxi Province (Grant No. 20202BABL201008).

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Correspondence to Zujin Zhang.

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Zhang, Z., Zhang, Y. On regularity criteria for the Navier–Stokes equations based on one directional derivative of the velocity or one diagonal entry of the velocity gradient. Z. Angew. Math. Phys. 72, 24 (2021). https://doi.org/10.1007/s00033-020-01442-1

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Keywords

  • Regularity criteria
  • Navier–Stokes equations
  • Regularity of solutions.

Mathematics Subject Classification

  • 35B65
  • 35Q30
  • 76D03