# American Institute of Mathematical Sciences

July & August  2021, 20(7&8): 2765-2787. doi: 10.3934/cpaa.2020244

## Remark on 3-D Navier-Stokes system with strong dissipation in one direction

 1 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 2 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author

Received  April 2020 Revised  July 2020 Published  July & August 2021 Early access  September 2020

Fund Project: The second author is supported by NSF of China under Grants 11688101 and 11371347, and innovation grant from National Center for Mathematics and Interdisciplinary Sciences

In this paper, we consider 3D anisotropic incompressible Navier-Stokes equations with strong dissipation in the vertical direction. We shall prove that this system has a unique global strong solution and the norm of the vertical component of the velocity field can be controlled by the norm of the corresponding component to the initial data. Similar result can also be obtained for the horizontal components of the vorticity. In particular, we simplify our proofs to the well-posedness result in our previous paper [11, 13].

Citation: Yanlin Liu, Ping Zhang. Remark on 3-D Navier-Stokes system with strong dissipation in one direction. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2765-2787. doi: 10.3934/cpaa.2020244
##### References:
 [1] J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.  doi: 10.1016/j.jfa.2008.07.008. [2] T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation, Ann. Math., 189 (2019), 101-144.  doi: 10.4007/annals.2019.189.1.3. [3] M. Cannone, Y. Meyer and F. Planchon, Solutions autosimilaires des équations de Navier-Stokes, Séminaire "Équations aux Dérivées Partielles" de l'École polytechnique, Exposé VIII, 1993–1994. doi: 10.1108/09533239410052824. [4] J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics, Clarendon Press, Oxford University Press, Oxford, 2006. [5] J. Y. Chemin and P. Zhang, On the critical one component regularity for 3-D Navier-Stokes system, Ann. Sci. Éc. Norm. Supér., 49 (2016), 131-167.  doi: 10.24033/asens.2278. [6] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188. [7] T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathop{\mathbb R\kern 0pt}\nolimits^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182. [8] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937. [9] O. A. Ladyzhenskaja, On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations, Zap. Nauchn. Sem. LOMI, 5 (1967), 169-185. [10] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354. [11] Y. Liu and P. Zhang, Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient, arXiv: 1708.04731. [12] M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Commun. Math. Phys., 307 (2011), 713-759.  doi: 10.1007/s00220-011-1350-6. [13] M. Paicu and P. Zhang, Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction, Sci. China Math., 62 (2019), 1175-1204.  doi: 10.1007/s11425-018-9504-1. [14] G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pure. Appl., 48 (1959), 173-182.  doi: 10.1007/BF02410664. [15] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. R at.Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344. [16] J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Probl., (1962), 69–98. [17] M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24. [18] T. Zhang, Erratum to: Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Commun. Math. Phys., 295 (2010), 877-884.  doi: 10.1007/s00220-010-1004-0.

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##### References:
 [1] J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.  doi: 10.1016/j.jfa.2008.07.008. [2] T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation, Ann. Math., 189 (2019), 101-144.  doi: 10.4007/annals.2019.189.1.3. [3] M. Cannone, Y. Meyer and F. Planchon, Solutions autosimilaires des équations de Navier-Stokes, Séminaire "Équations aux Dérivées Partielles" de l'École polytechnique, Exposé VIII, 1993–1994. doi: 10.1108/09533239410052824. [4] J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics, Clarendon Press, Oxford University Press, Oxford, 2006. [5] J. Y. Chemin and P. Zhang, On the critical one component regularity for 3-D Navier-Stokes system, Ann. Sci. Éc. Norm. Supér., 49 (2016), 131-167.  doi: 10.24033/asens.2278. [6] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188. [7] T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathop{\mathbb R\kern 0pt}\nolimits^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182. [8] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937. [9] O. A. Ladyzhenskaja, On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations, Zap. Nauchn. Sem. LOMI, 5 (1967), 169-185. [10] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354. [11] Y. Liu and P. Zhang, Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient, arXiv: 1708.04731. [12] M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Commun. Math. Phys., 307 (2011), 713-759.  doi: 10.1007/s00220-011-1350-6. [13] M. Paicu and P. Zhang, Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction, Sci. China Math., 62 (2019), 1175-1204.  doi: 10.1007/s11425-018-9504-1. [14] G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pure. Appl., 48 (1959), 173-182.  doi: 10.1007/BF02410664. [15] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. R at.Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344. [16] J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Probl., (1962), 69–98. [17] M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24. [18] T. Zhang, Erratum to: Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Commun. Math. Phys., 295 (2010), 877-884.  doi: 10.1007/s00220-010-1004-0.
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