• Previous Article
    Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains
  • CPAA Home
  • This Issue
  • Next Article
    A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain
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  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 & Applied Analysis, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics, Clarendon Press, Oxford University Press, Oxford, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

O. A. Ladyzhenskaja, On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations, Zap. Nauchn. Sem. LOMI, 5 (1967), 169-185.   Google Scholar

[10]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[11]

Y. Liu and P. Zhang, Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient, arXiv: 1708.04731. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Probl., (1962), 69–98.  Google Scholar

[17]

M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24.   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics, Clarendon Press, Oxford University Press, Oxford, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

O. A. Ladyzhenskaja, On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations, Zap. Nauchn. Sem. LOMI, 5 (1967), 169-185.   Google Scholar

[10]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[11]

Y. Liu and P. Zhang, Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient, arXiv: 1708.04731. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Probl., (1962), 69–98.  Google Scholar

[17]

M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24.   Google Scholar

[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.  Google Scholar

[1]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101

[2]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[3]

Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845

[4]

Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517

[5]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195

[6]

Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143

[7]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1

[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 & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148

[9]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020142

[10]

Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158

[11]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315

[12]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197

[13]

Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1

[14]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078

[15]

Adalet Hanachi, Haroune Houamed, Mohamed Zerguine. On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6473-6506. doi: 10.3934/dcds.2020287

[16]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437

[17]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053

[18]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056

[19]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5801-5815. doi: 10.3934/dcds.2016055

[20]

Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 901-916. doi: 10.3934/dcdss.2014.7.901

2019 Impact Factor: 1.105

Article outline

[Back to Top]