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Direction of Vorticity and a Refined Regularity Criterion for the Navier–Stokes Equations with Fractional Laplacian

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Abstract

We give a refined regularity criterion for solutions of the three-dimensional Navier–Stokes equations with fractional dissipative term \((-\Delta )^{\alpha /2}v\). The criterion is composed by the direction field of the vorticity and its magnitude simultaneously. Our result is a generalized of previous results by Beirão da Veiga and Berselli (Differ Integral Equ 15(3):345–356, 2002), and Zhou (ANZIAM J 46(3):309–316, 2005, Monatsh Math 144(3):251–257, 2005). Moreover, our result mentioned about the relation between the solution of the Navier–Stokes equations and the Euler equations.

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Acknowledgements

The author is grateful to Professor Yong Zhou for telling me information. He also thanks Professor Tsuyoshi Yoneda for kind help for his research on the Navier–Stokes equations, and Professor Takahito Kashiwabara for useful comments. He was supported by the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics (FMSP) at the University of Tokyo.

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  1. University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8941, Japan

    Kengo Nakai

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Communicated by Y. Giga

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Nakai, K. Direction of Vorticity and a Refined Regularity Criterion for the Navier–Stokes Equations with Fractional Laplacian. J. Math. Fluid Mech. 21, 21 (2019). https://doi.org/10.1007/s00021-019-0422-9

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