×

zbMATH — the first resource for mathematics

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity. (English) Zbl 1193.35144
Summary: This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the \(L^r\)-norm of the vertical velocity \(v\) for any \(1<r<\infty \) is globally bounded and that the \(L^{\infty }\)-norm of \(v\) controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace \((-\Delta )\delta \) for \(\delta >0\) would guarantee the global regularity of classical solutions.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76R50 Diffusion
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abidi, H.; Hmidi, T., On the global well-posedness for Boussinesq system, J. differential equations, 233, 199-220, (2007) · Zbl 1111.35032
[2] Cannon, J.R.; DiBenedetto, E., The initial value problem for the Boussinesq equations with data in \(L^p\), (), 129-144
[3] Cao, C.; Wu, J., Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, (19 January 2009)
[4] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. math., 203, 497-513, (2006) · Zbl 1100.35084
[5] Danchin, R.; Paicu, M., Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces, Phys. D, 237, 1444-1460, (2008) · Zbl 1143.76432
[6] Danchin, R.; Paicu, M., Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data, Comm. math. phys., 290, 1-14, (2009) · Zbl 1186.35157
[7] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, (19 September 2008)
[8] 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
[9] 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
[10] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for euler – boussinesq system with critical dissipation, (22 March 2009)
[11] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for a boussinesq – navier – stokes system with critical dissipation, (9 April 2009)
[12] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete contin. dyn. syst., 12, 1-12, (2005) · Zbl 1274.76185
[13] Majda, A.J.; Grote, M.J., Model dynamics and vertical collapse in decaying strongly stratified flows, Phys. fluids, 9, 2932-2940, (1997) · Zbl 1185.76764
[14] Majda, A.J., Introduction to PDEs and waves for the atmosphere and Ocean, Courant lect. notes math., vol. 9, (2003), AMS/CIMS · Zbl 1278.76004
[15] Miao, C.; Xue, L., On the global well-posedness of a class of boussinesq – navier – stokes systems, (2 October 2009)
[16] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York · Zbl 0713.76005
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.