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

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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

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

References:

[1]	R. A. Adams and J. F. Fournier, Sobolev Spaces,, 2nd ed., (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. 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. 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. 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. doi: 10.1016/j.jde.2011.05.027. Google Scholar
[6]	H. Amann, Linear and Quasilinear Parabolic Problems,, vol. I, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar
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[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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[16]	P. G. Drazin and W. H. Reid, Hydrodynamic Stability,, Cambridge University Press, (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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[29]	A. Lunardi, Interpolation Theory,, 2nd ed., (2009). Google Scholar
[30]	A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lecture Notes in Mathematics, (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. doi: 10.1086/147463. Google Scholar
[32]	J. Pedlosky, Geophysical Fluid Dyanmics,, Springer-Verlag, (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. 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. doi: 10.1016/j.aim.2011.05.008. Google Scholar

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References:

[1]	R. A. Adams and J. F. Fournier, Sobolev Spaces,, 2nd ed., (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. 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. 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. 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. doi: 10.1016/j.jde.2011.05.027. Google Scholar
[6]	H. Amann, Linear and Quasilinear Parabolic Problems,, vol. I, (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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[16]	P. G. Drazin and W. H. Reid, Hydrodynamic Stability,, Cambridge University Press, (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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[29]	A. Lunardi, Interpolation Theory,, 2nd ed., (2009). Google Scholar
[30]	A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean,, Courant Lecture Notes in Mathematics, (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. doi: 10.1086/147463. Google Scholar
[32]	J. Pedlosky, Geophysical Fluid Dyanmics,, Springer-Verlag, (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. 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. doi: 10.1016/j.aim.2011.05.008. Google Scholar

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