zbMATH — the first resource for mathematics

On the global well-posedness of 2-D Boussinesq system with variable viscosity. (English) Zbl 1353.35220
Summary: In this paper, we investigate the global well-posedness of 2-D Boussinesq system, which has variable kinematic viscosity and with thermal conductivity of \(| D | \theta\), with general initial data provided that the viscosity coefficient is sufficiently close to some positive constant in \(L^\infty\) norm.

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
42B25 Maximal functions, Littlewood-Paley theory
Full Text: DOI
[1] Abidi, H., Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Rev. Mat. Iberoam., 23, 2, 537-586, (2007) · Zbl 1175.35099
[2] Abidi, H., Sur l’unicité pour le système de Boussinesq avec diffusion non linéaire, J. Math. Pures Appl., 91, 9, 80-99, (2009) · Zbl 1156.35074
[3] Abidi, H.; Hmidi, T., On the global well-posedness for Boussinesq system, J. Differential Equations, 233, 199-220, (2007) · Zbl 1111.35032
[4] Abidi, H.; Zhang, P., On the well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity, J. Differential Equations, 259, 3755-3802, (2015) · Zbl 1320.35252
[5] Abidi, H.; Zhang, P., On the global well-posedness of 3-D density-dependent Navier-Stokes system with variable viscosity, Sci. China, 58, 1129-1150, (2015) · Zbl 1327.35294
[6] H. Abidi, P. Zhang, On the global well-posedness of 3-D Boussinesq system with variable viscosity, preprint, 2013.
[7] Bahouri, H.; Chemin, J. Y.; Danchin, R., Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften, (2011), Springer · Zbl 1227.35004
[8] Bony, J. M., Calcul symbolique et propagation des singularités pour LES équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér., 14, 209-246, (1981) · Zbl 0495.35024
[9] Cao, C.; Wu, J., Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208, 985-1004, (2013) · Zbl 1284.35140
[10] Chae, D., Global regularity for the 2-D Boussinesq equations with partial viscous terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084
[11] Coifman, R.; Lions, P. L.; Meyer, Y.; Semmes, S., Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72, 247-286, (1993) · Zbl 0864.42009
[12] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249, 511-528, (2004) · Zbl 1309.76026
[13] Córdoba, A.; Martínez, Á. D., A pointwise inequality for fractional Laplacians, Adv. Math., 280, 79-85, (2015) · Zbl 1323.35195
[14] Danchin, R., Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133, 1311-1334, (2003) · Zbl 1050.76013
[15] 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
[16] 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
[17] Desjardins, B., Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Ration. Mech. Anal., 137, 135-158, (1997) · Zbl 0880.76090
[18] Díaz, J. I.; Galiano, G., Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11, 59-82, (1998) · Zbl 0916.35087
[19] Gontsharowa, O., About the uniqueness of the solution of the two-dimensional non-stationary problem for the equations of free convection with viscosity depending on temperature, Sib. Math. J., 92, (1990)
[20] Hmidi, T., On a maximum principle and its application to logarithmically critical Boussinesq system, Anal. PDE, 4, 247-284, (2011) · Zbl 1264.35173
[21] Hmidi, T.; Keraani, S., On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12, 461-480, (2007) · Zbl 1154.35073
[22] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249, 2147-2174, (2010) · Zbl 1200.35228
[23] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36, 420-445, (2011) · Zbl 1284.76089
[24] Hou, T. Y.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185
[25] Huang, J.; Paicu, M., Decay estimates of global solutions to 2D incompressible inhomogeneous Navier-Stokes equations with variable viscosity, Discrete Contin. Dyn. Syst., 34, 4647-4669, (2014) · Zbl 1304.35492
[26] Pedloski, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York
[27] Rodrigues, J. F., Weak solutions for thermoconvective flows of Boussinesq-Stefan type, (Mathematical Topics in Fluid Mechanics, Lisbon, 1991, Pitman Res. Notes Math. Ser., vol. 274, (1992), Longman Sci. Tech. Harlow), 93-116 · Zbl 0794.76087
[28] Schonbek, M. E., Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11, 733-763, (1986) · Zbl 0607.35071
[29] 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
[30] Wiegner, M., Decay results for weak solutions to the Navier-Stokes equations on \(\mathbf{R}^n\), J. Lond. Math. Soc. (2), 35, 303-313, (1987) · Zbl 0652.35095
[31] Wu, J., Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys., 263, 803-831, (2006) · Zbl 1104.35037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.