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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

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.
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

[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

show all references

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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