Global regularity results for the 2D Boussinesq equations with vertical dissipation. (English) Zbl 1232.35111

Adhikari, Dhanapati; Cao, Chongsheng; Wu, Jiahong

J. Differ. Equations 251, No. 6, 1637-1655 (2011).

Authors’ abstract: This paper furthers the study of the authors [J. Differ. Equations 249, No. 5, 1078–1088 (2010; Zbl 1193.35144)] on the global regularity issue concerning the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. It is shown here that the vertical velocity \(v\) of any classical solution in the Lebesgue space \(L^q\) with \(2\leq q<\infty\) is bounded by \(C_1q\) for \(C_1\) independent of \(q\). This bound significantly improves the previous exponential bound. In addition, we prove that, if \(v\) satisfies \[ \int^T_0 \sup_{q\geq 2} {\| v(\cdot,t)\|^2_{L^q}\over q}\,dt< \infty, \] then the associated solution of the 2D Boussinesq equations preserves its smoothness on \([0, T]\). In particular, \(|v|_{L^q}\leq C_2\sqrt{q}\) implies global regularity.

Reviewer: A. D. Osborne (Keele)

MSC:

35Q30	Navier-Stokes equations
35B45	A priori estimates in context of PDEs
35B65	Smoothness and regularity of solutions to PDEs
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D09	Viscous-inviscid interaction
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Keywords:

2D Boussinesq equation; global regularity; vertical diffusion

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]	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]	D. Adhikari, R. Sharma, J. Wu, Numerical results on the 2D Boussinesq equations with vertical dissipation, in preparation.
[4]	Bergh, J.; Löfström, J., Interpolation spaces, an introduction, (1976), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0344.46071
[5]	Cannon, J.R.; DiBenedetto, E., The initial value problem for the Boussinesq equations with data in \(L^p\), (), 129-144
[6]	C. Cao, J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., in press. · Zbl 1213.35159
[7]	Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. math., 203, 497-513, (2006) · Zbl 1100.35084
[8]	Chemin, J.-Y., Perfect incompressible fluids, Oxford lecture ser. math. appl., vol. 14, (1998), Oxford Science Publications, Oxford University Press
[9]	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
[10]	Danchin, R.; Paicu, M., Global well-posedness issues for the inviscid Boussinesq system with yudovichʼs type data, Comm. math. phys., 290, 1-14, (2009) · Zbl 1186.35157
[11]	Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, (19 September 2008)
[12]	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
[13]	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
[14]	Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, (22 March 2009)
[15]	Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, (9 April 2009)
[16]	Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete contin. dyn. syst., 12, 1-12, (2005) · Zbl 1274.76185
[17]	Lai, M.J.; Pan, R.H.; Zhao, K., Initial boundary value problem for 2D viscous Boussinesq equations, Arch. ration. mech. anal., 199, 739-760, (2011) · Zbl 1231.35171
[18]	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
[19]	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
[20]	Miao, C.; Xue, L., On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, (2 October 2009)
[21]	Pedlosky, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York · Zbl 0713.76005
[22]	Zhao, K., 2D inviscid heat conductive Boussinesq system in a bounded domain, Michigan math. J., 59, 329-352, (2010) · Zbl 1205.35048