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Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces
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 |
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. |
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. |
[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. |
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