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Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity. (English) Zbl 1284.35322
Summary: We prove the global existence of strong solution to the initial-boundary value problem of the 2-D Boussinesq system and 3-D infinite Prandtl number model with viscosity and thermal conductivity depending on the temperature.

35Q30 Navier-Stokes equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35Q35 PDEs in connection with fluid mechanics
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
Full Text: DOI
[1] Abidi, H., Sur lʼunicité pour le système de Boussinesq avec diffusion non linéaire, J. Math. Pures Appl., 91, 80-99, (2009) · Zbl 1156.35074
[2] Adhikari, D.; Cao, C.; Wu, J., The 2D Boussinesq equations with vertical viscosity and vertical diffusivity, J. Differential Equations, 249, 1078-1088, (2010) · Zbl 1193.35144
[3] Adhikari, D.; Cao, C.; Wu, J., Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differential Equations, 251, 1637-1655, (2011) · Zbl 1232.35111
[4] Amann, H., Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Stud., 4, 417-430, (2004) · Zbl 1072.35103
[5] Cao, C.; Wu, J., Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation
[6] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084
[7] Danchin, R.; Paicu, M., LES théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136, 261-309, (2008) · Zbl 1162.35063
[8] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21, 421-457, (2011) · Zbl 1223.35249
[9] Desjardins, B., Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Ration. Mech. Anal., 137, 135-158, (1997) · Zbl 0880.76090
[10] Diaz, J. I.; Galiano, G., On the Boussinesq system with nonlinear thermal diffusion, Nonlinear Anal., 30, 3255-3263, (1997) · Zbl 0897.35065
[11] Gunzburger, M.; Saka, Y.; Wang, X. M., Well-posedness of the infinite Prandtl number model for convection with temperature-dependent viscosity, Anal. Appl., 7, 297-308, (2009) · Zbl 1169.35369
[12] Hmidi, T.; Keraani, S., On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58, 1591-1618, (2009) · Zbl 1178.35303
[13] Hmidi, T.; Rousset, F., Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 1227-1246, (2010) · Zbl 1200.35229
[14] Hmidi, T.; Rousset, F., Global well-posedness for the Euler-Boussinesq system with axisymmetric data, J. Funct. Anal., 260, 745-796, (2011) · Zbl 1220.35127
[15] Hou, T. Y.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185
[16] Krylov, N. V., Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal., 250, 521-558, (2007) · Zbl 1133.35052
[17] Lions, P.-L., Mathematical topics in fluid dynamics, vol. 1, incompressible models, Oxford Sci. Publ., (1996)
[18] Lorca, S. A.; Boldrini, J. L., The initial value problem for a generalized Boussinesq model, Nonlinear Anal., 36, 457-480, (1999) · Zbl 0930.35136
[19] Lorca, S. A.; Boldrini, J. L., The initial value problem for a generalized Boussinesq model: regularity and global existence of strong solutions, Mat. Contemp., 11, 71-94, (1996) · Zbl 0861.35080
[20] Ladyzhenskaya, O. A.; Solonnikov, V. A.; Uralʼtceva, N. N., Linear and quasi-linear parabolic equations, (1968), Amer. Math. Soc. Providence, RI
[21] Lai, M. J.; Pan, R. H.; Zhao, K., Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199, 739-760, (2011) · Zbl 1231.35171
[22] Lieberman, G. M., Second order parabolic differential equations, (1996), World Scientific Publishing Co., Inc. River Edge, NJ · Zbl 0884.35001
[23] Liskevich, V.; Zhang, Qi S., Extra regularity for parabolic equations with drift terms, Manuscripta Math., 113, 191-209, (2004) · Zbl 1063.35045
[24] Solonnikov, V. A., \(L^p\)-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci. (N. Y.), 105, 2448-2484, (2001)
[25] Turcotte, D. L.; Schubert, G., Geodynamics applications of continuum physics to geological problems, (1982), John Wiley and Sons
[26] Wang, C.; Zhang, Z., Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228, 43-62, (2011) · Zbl 1231.35180
[27] Wang, X. M., Infinite Prandtl number limit of Rayleigh-Bénard convection, Comm. Pure Appl. Math., 57, 1265-1282, (2004) · Zbl 1112.76032
[28] Wang, X. M., Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number, Comm. Pure Appl. Math., 61, 789-815, (2008) · Zbl 1143.35351
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