# 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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] P. Constantin, Euler and Navier-Stokes equations, Publ. Mat., 52 (2008), 235-265.  doi: 10.5565/PUBLMAT_52208_01. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. [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.

show all references

##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] P. Constantin, Euler and Navier-Stokes equations, Publ. Mat., 52 (2008), 235-265.  doi: 10.5565/PUBLMAT_52208_01. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. [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.
 [1] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [2] Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033 [3] Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151 [4] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [5] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [6] Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185 [7] Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837 [8] Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 [9] Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631 [10] Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas. Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations. Electronic Research Archive, 2021, 29 (3) : 2223-2247. doi: 10.3934/era.2020113 [11] Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic and Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545 [12] Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141 [13] Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228 [14] Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064 [15] Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299 [16] Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 [17] Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081 [18] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [19] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [20] Oleg Imanuvilov. On the asymptotic properties for stationary solutions to the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2301-2340. doi: 10.3934/dcds.2020366

2020 Impact Factor: 2.425