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

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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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  1. School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, 341000, Jiangxi, People’s Republic of China

    Zujin Zhang & Yali Zhang

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  1. Zujin Zhang
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  2. Yali 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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  • DOI: https://doi.org/10.1007/s00033-020-01442-1

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