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On Vortex Alignment and the Boundedness of the Lq-Norm of Vorticity in Incompressible Viscous Fluids

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Abstract

We show that the spatial Lq-norm (q > 5/3) of the vorticity of an incompressible viscous fluid in ℝ3 remains bounded uniformly in time, provided that the direction of vorticity is Hölder continuous in space, and that the space-time Lq-norm of vorticity is finite. The Hölder index depends only on q. This serves as a variant of the classical result by Constantin-Fefferman (Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. J. Math. 42 (1993), 775–789), and the related work by Grujić-Ruzmaikina (Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE, Indiana Univ. J. Math. 53 (2004), 1073–1080).

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Acknowledgements

The author is indebted to Professor Zhongmin Qian for many insightful discussions and generous sharing of ideas, to Professor Gui-Qiang Chen for his lasting support, and to Professor Zoran Grujić for communicating with us about the paper [16]. Part of this work was done during SL’s stay as a CRM-ISM postdoctoral fellow at the Centre de Recherches Mathématiques, Université de Montréal, and the Institut des Sciences Mathématiques. The author would like to thank these institutions for their hospitality.

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  1. Siran Li

    Present address: Current address: Department of Mathematics, New York University-Shanghai, office 1146, 1555 Century Avenue, Pudong, Shanghai, 200122, China

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  1. Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston, Texas, 77251, USA

    Siran Li

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Li, S. On Vortex Alignment and the Boundedness of the Lq-Norm of Vorticity in Incompressible Viscous Fluids. Acta Math Sci 40, 1700–1708 (2020). https://doi.org/10.1007/s10473-020-0606-7

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  • DOI: https://doi.org/10.1007/s10473-020-0606-7

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