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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). [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,, Discrete Continuous Dynam. Systems, 12 (2005), 1. [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. [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. [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. [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. [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. [23] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lect. Notes Math., (2003). [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. [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. [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. [27] J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). [28] Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Vol. 184,, Advances in Partial Differential Equations, (2008). [29] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations,, Comm. Partial Differential Equations, 18 (1993), 701. doi: 10.1080/03605309308820946. [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. [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. [32] X. Yang and Y. Qin, A regularity criteria for the 3D Boussinesq equations in Besov spaces,, preprint, (2011). [33] S. Zheng, Nonlinear Evolution Equations, Vol. 133,, Monographs and Surveys in Pure and Applied Mathematics, (2004). doi: 10.1201/9780203492222.

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic,, New York, (1975). [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,, Discrete Continuous Dynam. Systems, 12 (2005), 1. [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. [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. [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. [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. [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. [23] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lect. Notes Math., (2003). [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. [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. [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. [27] J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). [28] Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Vol. 184,, Advances in Partial Differential Equations, (2008). [29] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations,, Comm. Partial Differential Equations, 18 (1993), 701. doi: 10.1080/03605309308820946. [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. [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. [32] X. Yang and Y. Qin, A regularity criteria for the 3D Boussinesq equations in Besov spaces,, preprint, (2011). [33] S. Zheng, Nonlinear Evolution Equations, Vol. 133,, Monographs and Surveys in Pure and Applied Mathematics, (2004). doi: 10.1201/9780203492222.
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