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

Jishan Fan; Fucai Li; Gen  Nakamura

doi:10.3934/dcds.2016012

Article Contents

2016, Volume 36, Issue 9: 4915-4923. Doi: 10.3934/dcds.2016012

This issue Previous Article Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces Next Article Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach

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

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 2017

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D09.

Citation:

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[29]	A. Lunardi, Interpolation Theory, 2nd ed., Lecture Notes, Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009.
[30]	A. J. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003.
[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.
[32]	J. Pedlosky, Geophysical Fluid Dyanmics, Springer-Verlag, New York, 1987.
[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.
[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.