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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
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show all references

References:
[1]

2nd ed., Pure and Appl. Math. (Amsterdam), vol. 140, Amsterdam: Elsevier/Academic Press, 2003.  Google Scholar

[2]

J. Math. Pures Appl., 91 (2009), 80-99. doi: 10.1016/j.matpur.2008.09.004.  Google Scholar

[3]

J. Differential Equations, 233 (2007), 199-220. doi: 10.1016/j.jde.2006.10.008.  Google Scholar

[4]

J. Differential Equations, 249 (2010), 1078-1088. doi: 10.1016/j.jde.2010.03.021.  Google Scholar

[5]

J. Differential Equations, 251 (2011), 1637-1655. doi: 10.1016/j.jde.2011.05.027.  Google Scholar

[6]

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]

J. Math. Fluid Mech., 12 (2010), 397-411. doi: 10.1007/s00021-009-0295-4.  Google Scholar

[8]

Trans. Amer. Math. Soc., 364 (2012), 5057-5090. doi: 10.1090/S0002-9947-2012-05432-8.  Google Scholar

[9]

Adv. Math., 203 (2006), 497-513. doi: 10.1016/j.aim.2005.05.001.  Google Scholar

[10]

Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935-946. doi: 10.1017/S0308210500026810.  Google Scholar

[11]

Adv. Math., 230 (2012), 1618-1645. doi: 10.1016/j.aim.2012.04.004.  Google Scholar

[12]

Comm. Math. Phys., 290 (2009), 1-14. doi: 10.1007/s00220-009-0821-5.  Google Scholar

[13]

Math. Models Methods Appl. Sci., 21 (2011), 421-457. doi: 10.1142/S0218202511005106.  Google Scholar

[14]

Nonlinear Anal., 30 (1997), 3255-3263. doi: 10.1016/S0362-546X(97)00330-1.  Google Scholar

[15]

Topol. Methods Nonlinear Anal., 11 (1998), 59-82.  Google Scholar

[16]

Cambridge University Press, Cambridge, 1981.  Google Scholar

[17]

J. Math. Phys., 55 (2014), 051505, 14pp. doi: 10.1063/1.4878495.  Google Scholar

[18]

Nonlinear Anal., 75 (2012), 3436-3442. doi: 10.1016/j.na.2012.01.008.  Google Scholar

[19]

Nonlinearity, 22 (2009), 553-568. doi: 10.1088/0951-7715/22/3/003.  Google Scholar

[20]

Indiana Univ. Math. J., 58 (2009), 1591-1618. doi: 10.1512/iumj.2009.58.3590.  Google Scholar

[21]

Comm. Partial Differential Equations, 36 (2011), 420-445. doi: 10.1080/03605302.2010.518657.  Google Scholar

[22]

J. Differential Equations, 249 (2010), 2147-2174. doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[23]

J. Funct. Anal., 260 (2011), 745-796. doi: 10.1016/j.jfa.2010.10.012.  Google Scholar

[24]

Discrete Contin. Dyn. Syst., 12 (2005), 1-12. doi: 10.3934/dcds.2005.12.1.  Google Scholar

[25]

Arch. Ration. Mech. Anal., 199 (2011), 739-760. doi: 10.1007/s00205-010-0357-z.  Google Scholar

[26]

J. Hyperbolic Differ. Equ., 12 (2015), 469-488. doi: 10.1142/S0219891615500137.  Google Scholar

[27]

Nonlinear Anal., 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4.  Google Scholar

[28]

Mat. Contemp., 11 (1996), 71-94.  Google Scholar

[29]

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

[30]

Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003.  Google Scholar

[31]

Astron. J., 136 (1962), 1126-1133. doi: 10.1086/147463.  Google Scholar

[32]

Springer-Verlag, New York, 1987. Google Scholar

[33]

J. Differential Equations, 255 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[34]

Adv. Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.  Google Scholar

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