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September  2016, 36(9): 4915-4923. doi: 10.3934/dcds.2016012

Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain

Jishan Fan 1, , Fucai Li 2, and  Gen Nakamura 3,

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037
2. 

Department of Mathematics, Nanjing University, Nanjing 210093
3. 

Department of Mathematics, Hokkaido University, Sapporo, 060-0810

Received  June 2015 Revised  March 2016 Published  May 2016

In this paper we establish some regularity criteria for the three-dimensional Boussinesq system with the temperature-dependent viscosity and thermal diffusivity in a bounded domain.
Keywords: regularity criterion, temperature-dependent viscosity, thermal diffusivity, global strong solution., Boussinesq system.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D0.
Citation: Jishan Fan, Fucai Li, Gen Nakamura. Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4915-4923. doi: 10.3934/dcds.2016012
References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd ed., Pure and Appl. Math. (Amsterdam), vol. 140, Amsterdam: Elsevier/Academic Press, 2003.  Google Scholar

[2]

H. Abidi, Sur l'unicité pour le système de Boussinesq avec diffusion non linéaire, J. Math. Pures Appl., 91 (2009), 80-99. doi: 10.1016/j.matpur.2008.09.004.  Google Scholar

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D. Adhikari, C. Cao and J. Wu, Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differential Equations, 251 (2011), 1637-1655. doi: 10.1016/j.jde.2011.05.027.  Google Scholar

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H. Amann, Linear and Quasilinear Parabolic Problems, vol. I, volume 89 of Monographs in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

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H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions, An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411. doi: 10.1007/s00021-009-0295-4.  Google Scholar

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L. Brandolese and M. E. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Amer. Math. Soc., 364 (2012), 5057-5090. doi: 10.1090/S0002-9947-2012-05432-8.  Google Scholar

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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.  Google Scholar

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D. Chae and H.-S. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935-946. doi: 10.1017/S0308210500026810.  Google Scholar

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D. Chae and J. Wu, The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230 (2012), 1618-1645. doi: 10.1016/j.aim.2012.04.004.  Google Scholar

[12]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14. doi: 10.1007/s00220-009-0821-5.  Google Scholar

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R. Danchin and M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21 (2011), 421-457. doi: 10.1142/S0218202511005106.  Google Scholar

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J. I. Diaz and G. Galliano, On the Boussinesq system with nonlinear thermal diffusion, Nonlinear Anal., 30 (1997), 3255-3263. doi: 10.1016/S0362-546X(97)00330-1.  Google Scholar

[15]

J. I. Diaz and G. Galiano, Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11 (1998), 59-82.  Google Scholar

[16]

P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981.  Google Scholar

[17]

J.-S. Fan, F.-C. Li and G.Nakamura, Regularity criteria and uniform estimates for the Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, J. Math. Phys., 55 (2014), 051505, 14pp. doi: 10.1063/1.4878495.  Google Scholar

[18]

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), 3436-3442. doi: 10.1016/j.na.2012.01.008.  Google Scholar

[19]

J.-S. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568. doi: 10.1088/0951-7715/22/3/003.  Google Scholar

[20]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618. doi: 10.1512/iumj.2009.58.3590.  Google Scholar

[21]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445. doi: 10.1080/03605302.2010.518657.  Google Scholar

[22]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174. doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[23]

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.  Google Scholar

[24]

T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12. doi: 10.3934/dcds.2005.12.1.  Google Scholar

[25]

M.-J. Lai, R. Pan, K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760. doi: 10.1007/s00205-010-0357-z.  Google Scholar

[26]

H. Li, R. Pan and W. Zhang, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion, J. Hyperbolic Differ. Equ., 12 (2015), 469-488. doi: 10.1142/S0219891615500137.  Google Scholar

[27]

S. A. Lorca and J. L. Boldrini, The initial value problem for a generalized Boussinesq model, Nonlinear Anal., 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4.  Google Scholar

[28]

S. A. Lorca and J. L. Boldrini, The initial value problem for a generalized Boussinesq model: regularity and global existence of strong solutions, Mat. Contemp., 11 (1996), 71-94.  Google Scholar

[29]

A. Lunardi, Interpolation Theory, 2nd ed., Lecture Notes, Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009.  Google Scholar

[30]

A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003.  Google Scholar

[31]

J. M. Milhaljan, A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid, Astron. J., 136 (1962), 1126-1133. doi: 10.1086/147463.  Google Scholar

[32]

J. Pedlosky, Geophysical Fluid Dyanmics, Springer-Verlag, New York, 1987. Google Scholar

[33]

Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[34]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd ed., Pure and Appl. Math. (Amsterdam), vol. 140, Amsterdam: Elsevier/Academic Press, 2003.  Google Scholar

[2]

H. Abidi, Sur l'unicité pour le système de Boussinesq avec diffusion non linéaire, J. Math. Pures Appl., 91 (2009), 80-99. doi: 10.1016/j.matpur.2008.09.004.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

D. Adhikari, C. Cao and J. Wu, Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differential Equations, 251 (2011), 1637-1655. doi: 10.1016/j.jde.2011.05.027.  Google Scholar

[6]

H. Amann, Linear and Quasilinear Parabolic Problems, vol. I, volume 89 of Monographs in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[7]

H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions, An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411. doi: 10.1007/s00021-009-0295-4.  Google Scholar

[8]

L. Brandolese and M. E. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Amer. Math. Soc., 364 (2012), 5057-5090. doi: 10.1090/S0002-9947-2012-05432-8.  Google Scholar

[9]

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.  Google Scholar

[10]

D. Chae and H.-S. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935-946. doi: 10.1017/S0308210500026810.  Google Scholar

[11]

D. Chae and J. Wu, The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230 (2012), 1618-1645. doi: 10.1016/j.aim.2012.04.004.  Google Scholar

[12]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14. doi: 10.1007/s00220-009-0821-5.  Google Scholar

[13]

R. Danchin and M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21 (2011), 421-457. doi: 10.1142/S0218202511005106.  Google Scholar

[14]

J. I. Diaz and G. Galliano, On the Boussinesq system with nonlinear thermal diffusion, Nonlinear Anal., 30 (1997), 3255-3263. doi: 10.1016/S0362-546X(97)00330-1.  Google Scholar

[15]

J. I. Diaz and G. Galiano, Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11 (1998), 59-82.  Google Scholar

[16]

P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981.  Google Scholar

[17]

J.-S. Fan, F.-C. Li and G.Nakamura, Regularity criteria and uniform estimates for the Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, J. Math. Phys., 55 (2014), 051505, 14pp. doi: 10.1063/1.4878495.  Google Scholar

[18]

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), 3436-3442. doi: 10.1016/j.na.2012.01.008.  Google Scholar

[19]

J.-S. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568. doi: 10.1088/0951-7715/22/3/003.  Google Scholar

[20]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618. doi: 10.1512/iumj.2009.58.3590.  Google Scholar

[21]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445. doi: 10.1080/03605302.2010.518657.  Google Scholar

[22]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174. doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[23]

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.  Google Scholar

[24]

T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12. doi: 10.3934/dcds.2005.12.1.  Google Scholar

[25]

M.-J. Lai, R. Pan, K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760. doi: 10.1007/s00205-010-0357-z.  Google Scholar

[26]

H. Li, R. Pan and W. Zhang, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion, J. Hyperbolic Differ. Equ., 12 (2015), 469-488. doi: 10.1142/S0219891615500137.  Google Scholar

[27]

S. A. Lorca and J. L. Boldrini, The initial value problem for a generalized Boussinesq model, Nonlinear Anal., 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4.  Google Scholar

[28]

S. A. Lorca and J. L. Boldrini, The initial value problem for a generalized Boussinesq model: regularity and global existence of strong solutions, Mat. Contemp., 11 (1996), 71-94.  Google Scholar

[29]

A. Lunardi, Interpolation Theory, 2nd ed., Lecture Notes, Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009.  Google Scholar

[30]

A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003.  Google Scholar

[31]

J. M. Milhaljan, A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid, Astron. J., 136 (1962), 1126-1133. doi: 10.1086/147463.  Google Scholar

[32]

J. Pedlosky, Geophysical Fluid Dyanmics, Springer-Verlag, New York, 1987. Google Scholar

[33]

Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[34]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.  Google Scholar

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