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Local Regularity Criteria in Terms of One Velocity Component for the Navier–Stokes Equations

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Abstract

This paper is devoted to presenting new interior regularity criteria in terms of one velocity component for weak solutions to the Navier–Stokes equations in three dimensions. It is shown that the velocity is regular near a point z if its scaled \(L^p_tL^q_x\)-norm of some quantities related to the velocity field is finite and the scaled \(L^p_tL^q_x\)-norm of one velocity component is sufficiently small near z.

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Notes

  1. That means \(\Vert v_\lambda \Vert _{L^p \big (0,\lambda ^{-2}T;{\dot{H}}^{\frac{1}{2} + \frac{2}{p}}({\mathbb {R}}^3) \big )} = \Vert v\Vert _{L^p \big (0,T;{\dot{H}}^{\frac{1}{2} + \frac{2}{p}}({\mathbb {R}}^3) \big )}\) for \(v_\lambda (x,t) = \lambda v(\lambda x,\lambda ^2 t)\) with \(\lambda > 0\).

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Acknowledgements

K. Kang was supported by NRF-2019R1A2C1084685. D. D. Nguyen was supported by NRF-2015R1A5A1009350.

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  1. Department of Mathematics, Yonsei University, 50 Yonsei-ro Seodaemun-gu, 03722, Seoul, Republic of Korea

    Kyungkeun Kang & Dinh Duong Nguyen

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  1. Kyungkeun Kang
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Kang, K., Nguyen, D.D. Local Regularity Criteria in Terms of One Velocity Component for the Navier–Stokes Equations. J. Math. Fluid Mech. 25, 10 (2023). https://doi.org/10.1007/s00021-022-00754-8

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