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A Regularity Criterion for the Navier-Stokes Equations in Terms of One Directional Derivative of the Velocity

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Abstract

We consider the regularity criterion for the three dimensional incompressible Navier-Stokes equations in terms of one directional derivative of the velocity. The result shows that if weak solution u satisfies

$$\begin{aligned} \partial_{3}u\in L^{\frac{2}{1-r}}\bigl(0,T;\dot{M}^{p,\frac{3}{r}}\bigl( \mathbb{R}^{3}\bigr)\bigr) \end{aligned}$$

with 0<r<1 and \(2\leq p\leq\frac{3}{r}\), then u is regular on \((0,T]\times\mathbb{R}^{3}\). Here, \(\dot{M}^{p,\frac{3}{r}}(\mathbb{R}^{3})\) is the homogeneous Morrey-Campanato space.

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Notes

  1. In paper [5], Cao and Wu obtained that if \(\partial_{3} u\in L^{\frac{2}{1-r}}(0,T;L^{\frac{3}{r}}(\mathbb{R}^{3}))\) with 0≤r<1, then the solutions of the MHD equations is smooth on (0,T]. It is easy to verify that their regularity result still holds for the NS equations. In fact, when we let \(p=\frac{3}{r}\) in (2.3), then there holds \(\dot{M}^{\frac{3}{r},\frac{3}{r}}(\mathbb{R}^{3})=L^{\frac {3}{r}}(\mathbb{R}^{3})\), and the assumption (2.3) becomes that of [5].

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Acknowledgements

The author would like to thank the anonymous referees for their careful reading of the manuscript, and for the valuable suggestions.

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  1. Department of Mathematics, Hunan Normal University, Changsha, Hunan, 410081, People’s Republic of China

    Qiao Liu

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Correspondence to Qiao Liu.

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The author is supported by the National Natural Science Foundation of China (11326155, 11401202), the Hunan Provincial Natural Science Foundation of China (13JJ4043), and the Scientific Research Fund of Hunan Provincial Education Department (14B117).

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Liu, Q. A Regularity Criterion for the Navier-Stokes Equations in Terms of One Directional Derivative of the Velocity. Acta Appl Math 140, 1–9 (2015). https://doi.org/10.1007/s10440-014-9975-z

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