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).
Siran LI
. ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE Lq-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS[J]. Acta mathematica scientia, Series B, 2020
, 40(6)
: 1700
-1708
.
DOI: 10.1007/s10473-020-0606-7
[1] Beirão da Veiga H, Berselli L C. On the regularizing effect of the vorticity direction in incompressible viscous flows. Differ Integr Equ, 2002, 15:345-356
[2] Beirão da Veiga H, Berselli L C. Navier-Stokes equations:Green's matrices, vorticity direction, and regularity up to the boundary. J Differ Equ, 2009, 246:597-628
[3] Beirão da Veiga H. Vorticity and smoothness in viscous flows//Nonlinear Problems in Mathematical Physics and Related Topics, II. Int Math Ser (N Y), 2. New York:Kluwer/Plenum, 2002:61-67
[4] Beirão da Veiga H. Direction of vorticity and regularity up to the boundary:on the Lipschitz-continuous case. J Math Fluid Mech, 2013, 15:55-63
[5] Beirão da Veiga H. Direction of vorticity and smoothness of viscous fluid flows subjected to boundary constraints. Acta Appl Math, 2014, 132:63-71
[6] Beirão da Veiga H. On a family of results concerning direction of vorticity and regularity for the Navier-Stokes equations. Ann Univ Ferrara, 2014, 60:20-34
[7] Berselli L C. Some geometric constraints and the problem of global regularity for the Navier-Stokes equations. Nonlinearity, 2009, 22:2561-2581
[8] Chae D. On the regularity conditions for the Navier-Stokes and related equations. Rev Math Iberoam, 2007, 23:371-384
[9] Chae D, Kang D, Lee J. On the interior regularity of suitable weak solutions to the Navier-Stokes equations. Commun Partial Differ Equ, 2007, 32:1189-1207
[10] Constantin P, Fefferman C L. Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ J Math, 1993, 42:775-789
[11] Constantin P, Fefferman C L, Majda A J. Geometric constraints on potentially singular solutions for the 3-D Euler equations. Commun Partial Differ Equ, 1996, 21:559-571
[12] Constantin P, Foias C. Navier-Stokes Equations. Chicago:University of Chicago Press, 1988
[13] Fefferman C L. Existence and smoothness of the Navier-Stokes equations. The Millennium Prize Problems, 2006:57-67. https://www.claymath.org/sites/default/files/navierstokes.pdf
[14] Giga Y, Miura H. On vorticity directions near singularities for the Navier-Stokes flows with infinite energy. Comm Math Phys, 2011, 303:289-300
[15] Grujić Z. Localization and geometric depletion of vortex-stretching in the 3D NSE. Comm Math Phys, 2009, 290:861-870
[16] Grujić Z, Ruzmaikina A. Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE. Indiana Univ Math J, 2004, 53:1073-1080
[17] Lieb E H, Loss M. Analysis. Volume 14 of Graduate Studies in Mathematics. Providence, RI:American Mathematical Society, 2001
[18] Nirenberg L. On elliptic partial differential equations. Ann Scuola Norm Sup Pisa, 1959, 13:115-162
[19] Prodi G. Un teorema di unicità per le equazioni di Navier-Stokes. Ann Mat Pura Appl, 1959, 48:173-182
[20] Qian Z. An estimate for the vorticity of the Navier-Stokes equation. C R Acad Sci Paris, Ser I, 2009, 347:89-92
[21] Serrin J. The Initial Value Problem for the Navier-Stokes Equations//Noloinear Problems. Madison:The University Wisconsin Press, 1963:69-83
[22] Seregin G. Lecture Notes on Regularity Theory for the Navier-Stokes Equations. World Scientific, 2015
[23] Vasseur A. Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity. Appl Math, 2009, 54:47-52
[24] Zhou Y. A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity. Monatsh Math, 2005, 144:251-257
