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Locally Space-Time Anisotropic Regularity Criteria for the Navier–Stokes Equations in Terms of Two Vorticity Components

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Abstract

In this paper we prove the regularity of Leray weak solutions of the Navier–Stokes equations as long as the vorticity projection to a plane is bounded in the scale critical space \(L^p(0,T;L^q)\), \(2/p+3/q=2\), \(q \in (3/2,\infty )\). The plane may vary in space and time while the unit vector \(v=v(x,t)\) orthogonal to the plane is locally a Hölder function in space with the coefficient 1/2. This extends previous works by Chae and Choe and by Miller. We further show that a generalized form of this criterion improves several other regularity criteria in terms of the vorticity direction known from the literature.

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Acknowledgements

The author was supported by the Grant Agency of the Czech republic through Grant 18-09628S and by the Czech Academy of Sciences through RVO: 67985874.

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  1. The Czech Academy of Sciences, Institute of Hydrodynamics, Pod Patankou 30/5, 166 12, Prague 6, Czech Republic

    Zdenek Skalak

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  1. Zdenek Skalak
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Communicated by G. P. Galdi.

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Skalak, Z. Locally Space-Time Anisotropic Regularity Criteria for the Navier–Stokes Equations in Terms of Two Vorticity Components. J. Math. Fluid Mech. 23, 41 (2021). https://doi.org/10.1007/s00021-020-00544-0

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  • DOI: https://doi.org/10.1007/s00021-020-00544-0

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