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On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid
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 and Continuous Dynamical Systems, 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), 1078-1088. 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), 1637-1655. 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), 199-220. 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), 737-756. 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), 985-1004. 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), 497-513. 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), 645-671. 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), Art. IDrnn016, 31pp. 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), 983-1012. 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., TMA, 75 (2012), 3436-3442. 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), 1591-1618. 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), 2147-2174. 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), 420-445. 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), 1227-1246. 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), 745-796. 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), 1-12. [18] T. Y. Hou and C. Li, Dynamic stability of 3D axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697. 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), 1622-1637. 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), 96-99. 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), 739-760. 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., TMA, 95 (2014), 580-591. 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., Vol. {9}, American Mathematical Society/CIMA, 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), 707-735. 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), 33-67. 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), NATO Sci. Ser. II Math. Phys. Chem., 47, Kluwer Academic Publishers, Dordrecht, 2001, 3-10. doi: 10.1007/978-94-010-0446-6\_1. [27] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [28] Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Vol. 184, Advances in Partial Differential Equations, Birkhaüser Verlag AG, Basel-Boston-Berlin, 2008. [29] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations, 18 (1993), 701-727. 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), 854-859. 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., RWA, 17 (2014), 137-146. 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, Chapman & Hall/CRC, 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), 1078-1088. 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), 1637-1655. 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), 199-220. 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), 737-756. 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), 985-1004. 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), 497-513. 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), 645-671. 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), Art. IDrnn016, 31pp. 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), 983-1012. 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., TMA, 75 (2012), 3436-3442. 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), 1591-1618. 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), 2147-2174. 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), 420-445. 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), 1227-1246. 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), 745-796. 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), 1-12. [18] T. Y. Hou and C. Li, Dynamic stability of 3D axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697. 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), 1622-1637. 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), 96-99. 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), 739-760. 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., TMA, 95 (2014), 580-591. 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., Vol. {9}, American Mathematical Society/CIMA, 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), 707-735. 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), 33-67. 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), NATO Sci. Ser. II Math. Phys. Chem., 47, Kluwer Academic Publishers, Dordrecht, 2001, 3-10. doi: 10.1007/978-94-010-0446-6\_1. [27] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [28] Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Vol. 184, Advances in Partial Differential Equations, Birkhaüser Verlag AG, Basel-Boston-Berlin, 2008. [29] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations, 18 (1993), 701-727. 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), 854-859. 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., RWA, 17 (2014), 137-146. 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, Chapman & Hall/CRC, 2004. doi: 10.1201/9780203492222.
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