ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE Lq-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS

doi:10.1007/s10473-020-0606-7

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (6): 1700-1708.doi: 10.1007/s10473-020-0606-7

ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE L^q-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS

李思然^1,2

1. Department of Mathematics, Rice University, MS 136 P. O. Box 1892, Houston, Texas, 77251, USA;
2. Current Address:Department of Mathematics, New York University-Shanghai, office 1146, 1555 Century Avenue, Pudong, Shanghai 200122, China

收稿日期:2019-08-15 修回日期:2019-12-09 出版日期:2020-12-25 发布日期:2020-12-30
作者简介:Siran LI,E-mail:Siran.Li@rice.edu

ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE L^q-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS

Siran LI^1,2

1. Department of Mathematics, Rice University, MS 136 P. O. Box 1892, Houston, Texas, 77251, USA;
2. Current Address:Department of Mathematics, New York University-Shanghai, office 1146, 1555 Century Avenue, Pudong, Shanghai 200122, China

Received:2019-08-15 Revised:2019-12-09 Online:2020-12-25 Published:2020-12-30

摘要/Abstract

摘要： We show that the spatial $L^q$-norm ($q>5/3$) of the vorticity of an incompressible viscous fluid in $\mathbb{R}^3$ remains bounded uniformly in time, provided that the direction of vorticity is Hölder continuous in space, and that the space-time $L^q$-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).

关键词: Navier-Stokes equations, vorticity, regularity, vortex alignment, weak solution, strong solution, incompressible fluid

Abstract: We show that the spatial $L^q$-norm ($q>5/3$) of the vorticity of an incompressible viscous fluid in $\mathbb{R}^3$ remains bounded uniformly in time, provided that the direction of vorticity is Hölder continuous in space, and that the space-time $L^q$-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).

Key words: Navier-Stokes equations, vorticity, regularity, vortex alignment, weak solution, strong solution, incompressible fluid

中图分类号:

35Q30

李思然. ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE L^q-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS[J]. 数学物理学报(英文版), 2020, 40(6): 1700-1708.

Siran LI. ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE L^q-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS[J]. Acta mathematica scientia,Series B, 2020, 40(6): 1700-1708.

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ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE L^q-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS

ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE L^q-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS

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