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March  2016, 36(3): 1563-1581. doi: 10.3934/dcds.2016.36.1563

Global existence of solutions for the three-dimensional Boussinesq system with anisotropic data

1. 

Department of Applied Mathematics, Donghua University, Shanghai 201620

2. 

College of Information Science and Technology, Donghua University, Shanghai 201620, China, China, China

Received  October 2014 Revised  April 2015 Published  August 2015

In this paper, we study the three-dimensional axisymmetric Boussinesq equations with swirl. We establish the global existence of solutions for the three-dimensional axisymmetric Boussinesq equations for a family of anisotropic initial data.
Citation: Yuming Qin, Yang Wang, Xing Su, Jianlin Zhang. Global existence of solutions for the three-dimensional Boussinesq system with anisotropic data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1563-1581. doi: 10.3934/dcds.2016.36.1563
References:
[1]

R. A. Adams, Sobolev Spaces, Academic,, New York, (1975). Google Scholar

[2]

D. Adhikari, C. Cao and J. Wu, The 2D Boussinesq equations with vertical viscosity and vertical diffusivity,, J. Differential Equations, 249 (2010), 1078. doi: 10.1016/j.jde.2010.03.021. Google Scholar

[3]

A. Adhikari, C. Cao and J. Wu, Global regularity results for the 2D Boussinesq equations with vertical disspation,, J. Differential Equations, 251 (2011), 1637. doi: 10.1016/j.jde.2011.05.027. Google Scholar

[4]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system,, J. Differential Equations, 233 (2007), 199. doi: 10.1016/j.jde.2006.10.008. Google Scholar

[5]

H. Abidi, T. Hmidi and K. Sahbi, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system,, Discrete Continuous Dynam. Systems - A, 29 (2011), 737. doi: 10.3934/dcds.2011.29.737. Google Scholar

[6]

J. Cao and J. Wu, Global regularity results for the 2D anisotropic Boussinesq equations with vertical dissipation,, Arch. Ration. Mech. Anal., 208 (2013), 985. doi: 10.1007/s00205-013-0610-3. Google Scholar

[7]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms,, Adv. Math., 203 (2006), 497. doi: 10.1016/j.aim.2005.05.001. Google Scholar

[8]

D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations,, Math. Z., 239 (2002), 645. doi: 10.1007/s002090100317. Google Scholar

[9]

C. C. Chen, R. M. Strain, T. P. Tsai and H. T. Yau, Lower bound on the blow-up rate of axisymmetric Navier-Stokes equations,, International Mathematics Reserch Notices, 9 (2008). doi: 10.1093/imrn/rnn016. Google Scholar

[10]

J. Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations,, Ann. Math., 173 (2011), 983. doi: 10.4007/annals.2011.173.2.9. Google Scholar

[11]

J. Fan, G. Nakamura and H. Wang, blow-up criteria of smooth solutions to the 3D Boussinesq system with zero viscosity in a bounded domain,, Nonlinear Anal., 75 (2012), 3436. doi: 10.1016/j.na.2012.01.008. Google Scholar

[12]

T. Hmidi and S. Keraani, On the global well-posedness for the Boussinesq system with zero viscosity,, Indiana Univ. Math. J., 58 (2009), 1591. doi: 10.1512/iumj.2009.58.3590. Google Scholar

[13]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Boussinesq-Navier-Stokes system with critical disspation,, J. Differential Equations, 249 (2010), 2147. doi: 10.1016/j.jde.2010.07.008. Google Scholar

[14]

T. Hmidi, S. Keraani and F. Rousset, Golbal well-posedness for Euler-Boussinesq system with critical disspation,, Comm. Partial Differential Equations, 36 (2011), 420. doi: 10.1080/03605302.2010.518657. Google Scholar

[15]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data,, Ann. I. Poincaŕe-AN., 27 (2010), 1227. doi: 10.1016/j.anihpc.2010.06.001. Google Scholar

[16]

T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data,, J. Funct. Anal., 260 (2011), 745. doi: 10.1016/j.jfa.2010.10.012. Google Scholar

[17]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,, Discrete Continuous Dynam. Systems, 12 (2005), 1. Google Scholar

[18]

T. Y. Hou and C. Li, Dynamic stability of 3D axisymmetric Navier-Stokes equations with swirl,, Comm. Pure Appl. Math., 61 (2008), 661. doi: 10.1002/cpa.20212. Google Scholar

[19]

T. Y. Hou, Z. Lei and C. Li, Global regularity of 3D axi-symmetric Navier-Stokes equations with anisotropic data,, Comm. Partial Differential Equations, 33 (2008), 1622. doi: 10.1080/03605300802108057. Google Scholar

[20]

L. Jin and J. Fan, Uniform regularity for the 2D Boussinesq system with a slip boundary condition,, J. Math. Anal. Appl., 400 (2013), 96. doi: 10.1016/j.jmaa.2012.10.051. Google Scholar

[21]

M. J. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations,, Arch. Rational Mech. Anal., 199 (2011), 739. doi: 10.1007/s00205-010-0357-z. Google Scholar

[22]

X. Liu and Y. Li, On the stability of global solutions to the 3D Boussinesq system,, Nonlinear Anal., 95 (2014), 580. doi: 10.1016/j.na.2013.10.011. Google Scholar

[23]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lect. Notes Math., (2003). Google Scholar

[24]

C. Miao and L. Xue, On the golbal well-posedness of a class of Boussinesq-Navier-Stokes systems,, Nonlinear Differential Equations Appl., 18 (2011), 707. doi: 10.1007/s00030-011-0114-5. Google Scholar

[25]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation,, Comm. Math. Phy., 321 (2013), 33. doi: 10.1007/s00220-013-1721-2. Google Scholar

[26]

H. K. Moffatt, Some remarks on topological fluids mechanics,, in An Introduction to the Geometry and Topology of Fulid Flows (ed. R. L. Ricca), (2001), 3. doi: 10.1007/978-94-010-0446-6\_1. Google Scholar

[27]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). Google Scholar

[28]

Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Vol. 184,, Advances in Partial Differential Equations, (2008). Google Scholar

[29]

W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations,, Comm. Partial Differential Equations, 18 (1993), 701. doi: 10.1080/03605309308820946. Google Scholar

[30]

X. Xu and Z. Ye, The lifespan of solutions to the inviscid 3D Boussinesq system,, Applied Mathematics Letters, 26 (2013), 854. doi: 10.1016/j.aml.2013.03.009. Google Scholar

[31]

F. Xu and J. Yuan, On the global well-posedness for the 2D Euler-Boussinesq system,, Nonlinear Anal., 17 (2014), 137. doi: 10.1016/j.nonrwa.2013.11.001. Google Scholar

[32]

X. Yang and Y. Qin, A regularity criteria for the 3D Boussinesq equations in Besov spaces,, preprint, (2011). Google Scholar

[33]

S. Zheng, Nonlinear Evolution Equations, Vol. 133,, Monographs and Surveys in Pure and Applied Mathematics, (2004). doi: 10.1201/9780203492222. Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic,, New York, (1975). Google Scholar

[2]

D. Adhikari, C. Cao and J. Wu, The 2D Boussinesq equations with vertical viscosity and vertical diffusivity,, J. Differential Equations, 249 (2010), 1078. doi: 10.1016/j.jde.2010.03.021. Google Scholar

[3]

A. Adhikari, C. Cao and J. Wu, Global regularity results for the 2D Boussinesq equations with vertical disspation,, J. Differential Equations, 251 (2011), 1637. doi: 10.1016/j.jde.2011.05.027. Google Scholar

[4]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system,, J. Differential Equations, 233 (2007), 199. doi: 10.1016/j.jde.2006.10.008. Google Scholar

[5]

H. Abidi, T. Hmidi and K. Sahbi, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system,, Discrete Continuous Dynam. Systems - A, 29 (2011), 737. doi: 10.3934/dcds.2011.29.737. Google Scholar

[6]

J. Cao and J. Wu, Global regularity results for the 2D anisotropic Boussinesq equations with vertical dissipation,, Arch. Ration. Mech. Anal., 208 (2013), 985. doi: 10.1007/s00205-013-0610-3. Google Scholar

[7]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms,, Adv. Math., 203 (2006), 497. doi: 10.1016/j.aim.2005.05.001. Google Scholar

[8]

D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations,, Math. Z., 239 (2002), 645. doi: 10.1007/s002090100317. Google Scholar

[9]

C. C. Chen, R. M. Strain, T. P. Tsai and H. T. Yau, Lower bound on the blow-up rate of axisymmetric Navier-Stokes equations,, International Mathematics Reserch Notices, 9 (2008). doi: 10.1093/imrn/rnn016. Google Scholar

[10]

J. Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations,, Ann. Math., 173 (2011), 983. doi: 10.4007/annals.2011.173.2.9. Google Scholar

[11]

J. Fan, G. Nakamura and H. Wang, blow-up criteria of smooth solutions to the 3D Boussinesq system with zero viscosity in a bounded domain,, Nonlinear Anal., 75 (2012), 3436. doi: 10.1016/j.na.2012.01.008. Google Scholar

[12]

T. Hmidi and S. Keraani, On the global well-posedness for the Boussinesq system with zero viscosity,, Indiana Univ. Math. J., 58 (2009), 1591. doi: 10.1512/iumj.2009.58.3590. Google Scholar

[13]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Boussinesq-Navier-Stokes system with critical disspation,, J. Differential Equations, 249 (2010), 2147. doi: 10.1016/j.jde.2010.07.008. Google Scholar

[14]

T. Hmidi, S. Keraani and F. Rousset, Golbal well-posedness for Euler-Boussinesq system with critical disspation,, Comm. Partial Differential Equations, 36 (2011), 420. doi: 10.1080/03605302.2010.518657. Google Scholar

[15]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data,, Ann. I. Poincaŕe-AN., 27 (2010), 1227. doi: 10.1016/j.anihpc.2010.06.001. Google Scholar

[16]

T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data,, J. Funct. Anal., 260 (2011), 745. doi: 10.1016/j.jfa.2010.10.012. Google Scholar

[17]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,, Discrete Continuous Dynam. Systems, 12 (2005), 1. Google Scholar

[18]

T. Y. Hou and C. Li, Dynamic stability of 3D axisymmetric Navier-Stokes equations with swirl,, Comm. Pure Appl. Math., 61 (2008), 661. doi: 10.1002/cpa.20212. Google Scholar

[19]

T. Y. Hou, Z. Lei and C. Li, Global regularity of 3D axi-symmetric Navier-Stokes equations with anisotropic data,, Comm. Partial Differential Equations, 33 (2008), 1622. doi: 10.1080/03605300802108057. Google Scholar

[20]

L. Jin and J. Fan, Uniform regularity for the 2D Boussinesq system with a slip boundary condition,, J. Math. Anal. Appl., 400 (2013), 96. doi: 10.1016/j.jmaa.2012.10.051. Google Scholar

[21]

M. J. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations,, Arch. Rational Mech. Anal., 199 (2011), 739. doi: 10.1007/s00205-010-0357-z. Google Scholar

[22]

X. Liu and Y. Li, On the stability of global solutions to the 3D Boussinesq system,, Nonlinear Anal., 95 (2014), 580. doi: 10.1016/j.na.2013.10.011. Google Scholar

[23]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lect. Notes Math., (2003). Google Scholar

[24]

C. Miao and L. Xue, On the golbal well-posedness of a class of Boussinesq-Navier-Stokes systems,, Nonlinear Differential Equations Appl., 18 (2011), 707. doi: 10.1007/s00030-011-0114-5. Google Scholar

[25]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation,, Comm. Math. Phy., 321 (2013), 33. doi: 10.1007/s00220-013-1721-2. Google Scholar

[26]

H. K. Moffatt, Some remarks on topological fluids mechanics,, in An Introduction to the Geometry and Topology of Fulid Flows (ed. R. L. Ricca), (2001), 3. doi: 10.1007/978-94-010-0446-6\_1. Google Scholar

[27]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). Google Scholar

[28]

Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Vol. 184,, Advances in Partial Differential Equations, (2008). Google Scholar

[29]

W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations,, Comm. Partial Differential Equations, 18 (1993), 701. doi: 10.1080/03605309308820946. Google Scholar

[30]

X. Xu and Z. Ye, The lifespan of solutions to the inviscid 3D Boussinesq system,, Applied Mathematics Letters, 26 (2013), 854. doi: 10.1016/j.aml.2013.03.009. Google Scholar

[31]

F. Xu and J. Yuan, On the global well-posedness for the 2D Euler-Boussinesq system,, Nonlinear Anal., 17 (2014), 137. doi: 10.1016/j.nonrwa.2013.11.001. Google Scholar

[32]

X. Yang and Y. Qin, A regularity criteria for the 3D Boussinesq equations in Besov spaces,, preprint, (2011). Google Scholar

[33]

S. Zheng, Nonlinear Evolution Equations, Vol. 133,, Monographs and Surveys in Pure and Applied Mathematics, (2004). doi: 10.1201/9780203492222. Google Scholar

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