# zbMATH — the first resource for mathematics

Global regularity results for the 2D Boussinesq equations with partial dissipation. (English) Zbl 1328.35161
Summary: The two-dimensional (2D) incompressible Boussinesq equations model geophysical fluids and play an important role in the study of the Raleigh-Bernard convection. Mathematically this 2D system retains some key features of the 3D Navier-Stokes and Euler equations such as the vortex stretching mechanism. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation or no dissipation at all. This paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases. These results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2D Boussinesq equations.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B65 Smoothness and regularity of solutions to PDEs 35Q85 PDEs in connection with astronomy and astrophysics 76W05 Magnetohydrodynamics and electrohydrodynamics 85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics 76R10 Free convection 35B45 A priori estimates in context of PDEs
Full Text:
##### References:
 [1] 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 [2] 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 [3] Adhikari, D.; Cao, C.; Wu, J.; Xu, X., Small global solutions to the damped two-dimensional Boussinesq equations, J. Differential Equations, 256, 3594-3613, (2014) · Zbl 1290.35193 [4] Cao, C.; Regmi, D.; Wu, J., The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differential Equations, 254, 2661-2681, (2013) · Zbl 1270.35143 [5] Cao, C.; Wu, J., Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208, 985-1004, (2013) · Zbl 1284.35140 [6] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084 [7] Choi, K.; Hou, T.; Kiselev, A.; Luo, G.; Sverak, V.; Yao, Y., On the finite-time blowup of a 1D model for the 3D axisymmetric Euler equations, (2015) [8] Choi, K.; Kiselev, A.; Yao, Y., Finite time blow up for a 1D model of 2D Boussinesq system, Comm. Math. Phys., 334, 1667-1679, (2015) · Zbl 1309.35072 [9] Constantin, P.; Doering, C. R., Infinite Prandtl number convection, J. Stat. Phys., 94, 159-172, (1999) · Zbl 0935.76083 [10] Constantin, P.; Vicol, V., Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22, 1289-1321, (2012) · Zbl 1256.35078 [11] Cui, X.; Dou, C.; Jiu, Q., Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces, J. Partial Differ. Equ., 25, 220-238, (2012) · Zbl 1274.35280 [12] Danchin, R., Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics, Proc. Amer. Math. Soc., 141, 1979-1993, (2013) · Zbl 1283.35080 [13] 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 [14] Doering, C.; Gibbon, J., Applied analysis of the Navier-Stokes equations, Cambridge Texts Appl. Math., (1995), Cambridge University Press Cambridge · Zbl 0838.76016 [15] Gill, A. E., Atmosphere-Ocean dynamics, (1982), Academic Press London [16] Hmidi, T., On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal. Partial Differential Equations, 4, 247-284, (2011) · Zbl 1264.35173 [17] 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 [18] 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 [19] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185 [20] Jiu, Q.; Miao, C.; Wu, J.; Zhang, Z., The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46, 3426-3454, (2014) · Zbl 1319.35193 [21] Jiu, Q.; Wu, J.; Yang, W., Eventual regularity of the two-dimensional Boussinesq equations with supercritical dissipation, J. Nonlinear Sci., 25, 37-58, (2015) · Zbl 1311.35221 [22] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 7, 891-907, (1988) · Zbl 0671.35066 [23] KC, D.; Regmi, D.; Tao, L.; Wu, J., The 2D Euler-Boussinesq equations with a singular velocity, J. Differential Equations, 257, 82-108, (2014) · Zbl 1291.35221 [24] Kenig, C. E.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized Korteweg-de-Vries equation via the contraction principle, Comm. Pure Appl. Math., 46, 527-620, (1993) · Zbl 0808.35128 [25] Lai, M.; Pan, R.; Zhao, K., Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199, 739-760, (2011) · Zbl 1231.35171 [26] Larios, A.; Lunasin, E.; Titi, E. S., Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255, 2636-2654, (2013) · Zbl 1284.35343 [27] Lei, Z.; Zhou, Y., BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25, 575-583, (2009) · Zbl 1171.35452 [28] Li, D.; Xu, X., Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity, Dyn. Partial Differ. Equ., 10, 255-265, (2013) · Zbl 1302.35319 [29] Luo, G.; Hou, T., Potentially singular solutions of the 3D incompressible Euler equations, (2013) [30] 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 [31] Majda, A. J.; Bertozzi, A. L., Vorticity and incompressible flow, (2001), Cambridge University Press Cambridge [32] Miao, C.; Xue, L., On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Appl., 18, 707-735, (2011) · Zbl 1235.76020 [33] Nirenberg, L., On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13, 115-162, (1959) · Zbl 0088.07601 [34] Ohkitani, K., Comparison between the Boussinesq and coupled Euler equations in two dimensions, (Tosio Kato’s Method and Principle for Evolution Equations Mathematical Physics, Sapporo, 2001, RIMS Kôkyûroku Bessatsu, vol. 1234, (2001)), 127-145 [35] Pedlosky, J., Geophysical fluid dyanmics, (1987), Springer-Verlag New York [36] Sarria, A.; Wu, J., Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations, (2014) [37] Wen, B.; Dianati, N.; Lunasin, E.; Chini, G.; Doering, C., New upper bounds and reduced dynamical modeling for Rayleigh-Bénard convection in a fluid saturated porous layer, Commun. Nonlinear Sci. Numer. Simul., 17, 2191-2199, (2012) [38] Wu, J., The 2D Boussinesq equations with partial or fractional dissipation, (Lin, Fang-Hua; Zhang, Ping, Lectures on the Analysis of Nonlinear Partial Differential Equations, Morningside Lect. Math., (2014), International Press Somerville, MA), in press [39] Wu, J.; Xu, X., Well-posedness and inviscid limits of the Boussinesq equations with fractional Laplacian dissipation, Nonlinearity, 27, 2215-2232, (2014) · Zbl 1301.35115 [40] Wu, J.; Xu, X.; Ye, Z., Global smooth solutions to the n-dimensional damped models of incompressible fluid mechanics with small initial datum, J. Nonlinear Sci., 25, 157-192, (2015) · Zbl 1311.35236 [41] Xu, X., Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Anal., 72, 677-681, (2010) · Zbl 1177.76024 [42] Yang, W.; Jiu, Q.; Wu, J., Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation, J. Differential Equations, 257, 4188-4213, (2014) · Zbl 1300.35108 [43] Ye, Z., Blow-up criterion of smooth solutions for the Boussinesq equations, Nonlinear Anal., 110, 97-103, (2014) · Zbl 1300.35109 [44] Ye, Z.; Xu, X., Remarks on global regularity of the 2D Boussinesq equations with fractional dissipation, Nonlinear Anal., 125, 715-724, (2015) · Zbl 1405.35171 [45] Zhao, K., 2D inviscid heat conductive Boussinesq equations on a bounded domain, Michigan Math. J., 59, 329-352, (2010) · Zbl 1205.35048
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.