# American Institute of Mathematical Sciences

April  2019, 12(2): 203-213. doi: 10.3934/dcdss.2019014

## Navier-Stokes equations: Some questions related to the direction of the vorticity

 Department of Mathematics, Pisa University, Italy, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

To Professor Vicentiu Rǎdulescu on the occasion of his 60th birthday

Received  June 2017 Revised  November 2017 Published  August 2018

Fund Project: Partially supported by FCT (Portugal) under grant UID/MAT/04561/3013.

We consider solutions $u$ to the Navier-Stokes equations in the whole space. We set $\omega = \nabla × u,$ the vorticity of $u$. Our study concerns relations between $\beta -$Hölder continuity assumptions on the direction of the vorticity and induced integrability regularity results, a significant research field starting from a pioneering 1993 paper by P. Constantin and Ch. Fefferman. Nowadays it is know that if $\beta = \frac{1}{2}$ then $\omega ∈ L^{∞}(L^2),$ a 2002 result by L.C. Berselli and the author. This conclusion implies smoothness of solutions. Assume now that one is able to prove that a strictly decreasing perturbation of $\beta$ near $\frac{1}{2}$ induces a strictly decreasing perturbation for $r$ near $2$. Since regularity holds if merely $\omega ∈ L^{∞}(L^r),$ for some $r≥ \frac32,$ the above assumption would imply regularity for values $\beta <\frac{1}{2}.$ The aim of the present note is to go deeper into this study and related open problems. The approach developed below reinforces the conjecture on the particular significance of the value $\beta = \frac{1}{2}.$

Citation: Hugo Beirão da Veiga. Navier-Stokes equations: Some questions related to the direction of the vorticity. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 203-213. doi: 10.3934/dcdss.2019014
##### References:
 [1] H. Beirão da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166.  doi: 10.1512/iumj.1987.36.36008.  Google Scholar [2] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbf{R}^n$, Chin. Ann. Math., Ser.B, 16 (1995), 407-412.   Google Scholar [3] H. Beirão da Veiga, Vorticity and smoothness in viscous flows, in Nonlinear Problems in Mathematical Physics and Related Topics, volume in Honor of O. A. Ladyzhenskaya, International Mathematical Series, Kluwer Academic, London, 2 (2002), 61–67. doi: 10.1007/978-1-4615-0701-7_3.  Google Scholar [4] H. Beirão da Veiga, Vorticity and regularity for flows under the Navier boundary condition, Comm. Pure Appl. Anal., 5 (2006), 907-918.  doi: 10.3934/cpaa.2006.5.907.  Google Scholar [5] H. Beirão da Veiga, Viscous incompressible flows under stress-free boundary conditions. The smoothness effect of near orthogonality or near parallelism between velocity and vorticity, Bollettino UMI, 5 (2012), 225-232.   Google Scholar [6] H. Beirão da Veiga, On a family of results concerning direction of vorticity and regularity for the Navier-Stokes equations, Ann. Univ. Ferrara, 60 (2014), 23-34.  doi: 10.1007/s11565-014-0206-3.  Google Scholar [7] H. Beirão da Veiga, Open problems concerning the Hőlder continuity of the direction of vorticity for the Navier-Stokes equations, arXiv: 1604.08083 [math. AP] 27 Apr 2016. Google Scholar [8] H. Beirão da Veiga and L. C. Berselli, On the regularizing effect of the vorticity direction in incompressible viscous flows, Differential Integral Equations, 15 (2002), 345-356.   Google Scholar [9] H. Beirão da Veiga and L. C. Berselli, Navier-Stokes equations: Green's matrices, vorticity direction, and regularity up to the boundary, J. Diff. Equations, 246 (2009), 597-628.  doi: 10.1016/j.jde.2008.02.043.  Google Scholar [10] L. C. Berselli, Some geometrical constraints and the problem of the global On regularity for the Navier-Stokes equations, Nonlinearity, 22 (2009), 2561-2581.  doi: 10.1088/0951-7715/22/10/013.  Google Scholar [11] L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 209-224.  doi: 10.1007/s11565-009-0076-2.  Google Scholar [12] L. C. Berselli and D. Córdoba, On the regularity of the solutions to the 3D Navier-Stokes equations: A remark on the role of helicity, C.R. Acad. Sci. Paris, Ser.I, 347 (2009), 613-618.  doi: 10.1016/j.crma.2009.03.003.  Google Scholar [13] D. Chae, On the regularity conditions for the Navier-Stokes and related equations, Rev. Mat. Iberoam., 23 (2007), 371-384.  doi: 10.4171/RMI/498.  Google Scholar [14] D. Chae, On the regularity conditions of suitable weak solutions of the 3D Navier-Stokes equations, J. Math. Fluid Mech., 12 (2010), 171-180.  doi: 10.1007/s00021-008-0280-3.  Google Scholar [15] D. Chae, K. Kang and J. Lee, On the interior regularity of suitable weak solutions to the Navier-Stokes equations, Comm. Part. Diff. Eq., 32 (2007), 1189-1207.  doi: 10.1080/03605300601088823.  Google Scholar [16] P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 603-621.  doi: 10.1090/S0273-0979-07-01184-6.  Google Scholar [17] P. Constantin, Euler and Navier-Stokes equations, Publ. Mat., 52 (2008), 235-265.  doi: 10.5565/PUBLMAT_52208_01.  Google Scholar [18] P. Constantin and Ch. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789.  doi: 10.1512/iumj.1993.42.42034.  Google Scholar [19] P. Constantin, Ch. Fefferman and A. Majda, Geometric constraints on potentially singular solutions for the 3D Euler equations, Comm. Partial Differ. Eq., 21 (1996), 559-571.  doi: 10.1080/03605309608821197.  Google Scholar [20] G.-H. Cottet, D. Jiroveanu and B. Michaux, Vorticity dynamics and turbulence models for large-eddy simulations, M2AN Math. Model. Numer. Anal., 37 (2003), 187-207.  doi: 10.1051/m2an:2003013.  Google Scholar [21] R. Dascaliuc and Z. Grujić, Coherent vortex structures and 3D enstrophy cascade, Comm. Math. Phys., 317 (2013), 547-561.  doi: 10.1007/s00220-012-1595-8.  Google Scholar [22] R. Dascaliuc and Z. Grujić, Vortex stretching and criticality for the three-dimensional Navier-Stokes equations, J. Math. Phys., 53 (2012), 115613, 9 pp. doi: 10.1063/1.4752170.  Google Scholar [23] L. Escauriaza, G. Seregin and V. Šverák, $L_{3, \, ∞}$-solutions to the Navier-Stokes equations and backward uniqueness, Russian Mathematical Surveys, 58 (2003), 211-250.  doi: 10.1070/RM2003v058n02ABEH000609.  Google Scholar [24] G. P. Galdi and P. Maremonti, Sulla regolarità delle soluzioni deboli al sistema di NavierStokes in domini arbitrari, Ann. Univ. Ferrara Sez. VII (N.S.), 34 (1988), 59-73.   Google Scholar [25] Y. Giga and H. Miura, On vorticity directions near singularities for the Navier-Stokes flows with infinite energy, Comm. Math. Phys., 303 (2011), 289-300.  doi: 10.1007/s00220-011-1197-x.  Google Scholar [26] Z. Grujić, Localization and geometric depletion of vortex-stretching in the 3D NSE, Comm. Math. Phys., 290 (2009), 861-870.  doi: 10.1007/s00220-008-0726-8.  Google Scholar [27] Z. Grujić and R. Guberović, Localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3D NSE, Comm. Math. Phys., 298 (2010), 407-418.  doi: 10.1007/s00220-010-1000-4.  Google Scholar [28] Z. Grujić and A. Ruzmaikina, Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE, Indiana Univ. Math. J., 53 (2004), 1073-1080.  doi: 10.1512/iumj.2004.53.2415.  Google Scholar [29] Z. Grujić and Q. S. Zhang, Space-time localization of a class of geometric criteria for preventing blow-up in the 3D NSE, Comm. Math. Phys., 262 (2006), 555-564.  doi: 10.1007/s00220-005-1437-z.  Google Scholar [30] N. Ju, Geometric depletion of vortex stretch in 3D viscous incompressible flow, J. Math. Anal. Appl, 321 (2006), 412-425.  doi: 10.1016/j.jmaa.2005.08.048.  Google Scholar [31] N. Ju, Geometric constrains for global regularity of 2D quasi-geostrophic flows, J. Differential Equations, 226 (2006), 54-79.  doi: 10.1016/j.jde.2006.03.010.  Google Scholar [32] A. Ruzmaikina and Z. Grujić, On depletion of the vortex-stretching term in the 3D Navier-Stokes equations, Comm. Math. Phys., 247 (2004), 601-611.  doi: 10.1007/s00220-004-1072-0.  Google Scholar [33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.  Google Scholar [34] A. Vasseur, Regularity criterion for $3D$ Navier-Stokes equations in terms of the direction of the velocity, Appl. Math., 54 (2009), 47-52.  doi: 10.1007/s10492-009-0003-y.  Google Scholar

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##### References:
 [1] H. Beirão da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166.  doi: 10.1512/iumj.1987.36.36008.  Google Scholar [2] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbf{R}^n$, Chin. Ann. Math., Ser.B, 16 (1995), 407-412.   Google Scholar [3] H. Beirão da Veiga, Vorticity and smoothness in viscous flows, in Nonlinear Problems in Mathematical Physics and Related Topics, volume in Honor of O. A. Ladyzhenskaya, International Mathematical Series, Kluwer Academic, London, 2 (2002), 61–67. doi: 10.1007/978-1-4615-0701-7_3.  Google Scholar [4] H. Beirão da Veiga, Vorticity and regularity for flows under the Navier boundary condition, Comm. Pure Appl. Anal., 5 (2006), 907-918.  doi: 10.3934/cpaa.2006.5.907.  Google Scholar [5] H. Beirão da Veiga, Viscous incompressible flows under stress-free boundary conditions. The smoothness effect of near orthogonality or near parallelism between velocity and vorticity, Bollettino UMI, 5 (2012), 225-232.   Google Scholar [6] H. Beirão da Veiga, On a family of results concerning direction of vorticity and regularity for the Navier-Stokes equations, Ann. Univ. Ferrara, 60 (2014), 23-34.  doi: 10.1007/s11565-014-0206-3.  Google Scholar [7] H. Beirão da Veiga, Open problems concerning the Hőlder continuity of the direction of vorticity for the Navier-Stokes equations, arXiv: 1604.08083 [math. AP] 27 Apr 2016. Google Scholar [8] H. Beirão da Veiga and L. C. Berselli, On the regularizing effect of the vorticity direction in incompressible viscous flows, Differential Integral Equations, 15 (2002), 345-356.   Google Scholar [9] H. Beirão da Veiga and L. C. Berselli, Navier-Stokes equations: Green's matrices, vorticity direction, and regularity up to the boundary, J. Diff. Equations, 246 (2009), 597-628.  doi: 10.1016/j.jde.2008.02.043.  Google Scholar [10] L. C. Berselli, Some geometrical constraints and the problem of the global On regularity for the Navier-Stokes equations, Nonlinearity, 22 (2009), 2561-2581.  doi: 10.1088/0951-7715/22/10/013.  Google Scholar [11] L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 209-224.  doi: 10.1007/s11565-009-0076-2.  Google Scholar [12] L. C. Berselli and D. Córdoba, On the regularity of the solutions to the 3D Navier-Stokes equations: A remark on the role of helicity, C.R. Acad. Sci. Paris, Ser.I, 347 (2009), 613-618.  doi: 10.1016/j.crma.2009.03.003.  Google Scholar [13] D. Chae, On the regularity conditions for the Navier-Stokes and related equations, Rev. Mat. Iberoam., 23 (2007), 371-384.  doi: 10.4171/RMI/498.  Google Scholar [14] D. Chae, On the regularity conditions of suitable weak solutions of the 3D Navier-Stokes equations, J. Math. Fluid Mech., 12 (2010), 171-180.  doi: 10.1007/s00021-008-0280-3.  Google Scholar [15] D. Chae, K. Kang and J. Lee, On the interior regularity of suitable weak solutions to the Navier-Stokes equations, Comm. Part. Diff. Eq., 32 (2007), 1189-1207.  doi: 10.1080/03605300601088823.  Google Scholar [16] P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 603-621.  doi: 10.1090/S0273-0979-07-01184-6.  Google Scholar [17] P. Constantin, Euler and Navier-Stokes equations, Publ. Mat., 52 (2008), 235-265.  doi: 10.5565/PUBLMAT_52208_01.  Google Scholar [18] P. Constantin and Ch. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789.  doi: 10.1512/iumj.1993.42.42034.  Google Scholar [19] P. Constantin, Ch. Fefferman and A. Majda, Geometric constraints on potentially singular solutions for the 3D Euler equations, Comm. Partial Differ. Eq., 21 (1996), 559-571.  doi: 10.1080/03605309608821197.  Google Scholar [20] G.-H. Cottet, D. Jiroveanu and B. Michaux, Vorticity dynamics and turbulence models for large-eddy simulations, M2AN Math. Model. Numer. Anal., 37 (2003), 187-207.  doi: 10.1051/m2an:2003013.  Google Scholar [21] R. Dascaliuc and Z. Grujić, Coherent vortex structures and 3D enstrophy cascade, Comm. Math. Phys., 317 (2013), 547-561.  doi: 10.1007/s00220-012-1595-8.  Google Scholar [22] R. Dascaliuc and Z. Grujić, Vortex stretching and criticality for the three-dimensional Navier-Stokes equations, J. Math. Phys., 53 (2012), 115613, 9 pp. doi: 10.1063/1.4752170.  Google Scholar [23] L. Escauriaza, G. Seregin and V. Šverák, $L_{3, \, ∞}$-solutions to the Navier-Stokes equations and backward uniqueness, Russian Mathematical Surveys, 58 (2003), 211-250.  doi: 10.1070/RM2003v058n02ABEH000609.  Google Scholar [24] G. P. Galdi and P. Maremonti, Sulla regolarità delle soluzioni deboli al sistema di NavierStokes in domini arbitrari, Ann. Univ. Ferrara Sez. VII (N.S.), 34 (1988), 59-73.   Google Scholar [25] Y. Giga and H. Miura, On vorticity directions near singularities for the Navier-Stokes flows with infinite energy, Comm. Math. Phys., 303 (2011), 289-300.  doi: 10.1007/s00220-011-1197-x.  Google Scholar [26] Z. Grujić, Localization and geometric depletion of vortex-stretching in the 3D NSE, Comm. Math. Phys., 290 (2009), 861-870.  doi: 10.1007/s00220-008-0726-8.  Google Scholar [27] Z. Grujić and R. Guberović, Localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3D NSE, Comm. Math. Phys., 298 (2010), 407-418.  doi: 10.1007/s00220-010-1000-4.  Google Scholar [28] Z. Grujić and A. Ruzmaikina, Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE, Indiana Univ. Math. J., 53 (2004), 1073-1080.  doi: 10.1512/iumj.2004.53.2415.  Google Scholar [29] Z. Grujić and Q. S. Zhang, Space-time localization of a class of geometric criteria for preventing blow-up in the 3D NSE, Comm. Math. Phys., 262 (2006), 555-564.  doi: 10.1007/s00220-005-1437-z.  Google Scholar [30] N. Ju, Geometric depletion of vortex stretch in 3D viscous incompressible flow, J. Math. Anal. Appl, 321 (2006), 412-425.  doi: 10.1016/j.jmaa.2005.08.048.  Google Scholar [31] N. Ju, Geometric constrains for global regularity of 2D quasi-geostrophic flows, J. Differential Equations, 226 (2006), 54-79.  doi: 10.1016/j.jde.2006.03.010.  Google Scholar [32] A. Ruzmaikina and Z. Grujić, On depletion of the vortex-stretching term in the 3D Navier-Stokes equations, Comm. Math. Phys., 247 (2004), 601-611.  doi: 10.1007/s00220-004-1072-0.  Google Scholar [33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.  Google Scholar [34] A. Vasseur, Regularity criterion for $3D$ Navier-Stokes equations in terms of the direction of the velocity, Appl. Math., 54 (2009), 47-52.  doi: 10.1007/s10492-009-0003-y.  Google Scholar
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