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Regularity criterion of the 2D Bénard equations with critical and supercritical dissipation. (English) Zbl 1367.35135
Summary: In this paper, we investigate the Cauchy problem for the two-dimensional (2D) incompressible Bénard equations. On the one hand, we prove global-in-time existence of smooth solutions to the 2D Bénard equations with critical dissipation. On the other hand, we establish several regularity criteria involving temperature for 2D Bénard equations with supercritical dissipation.

##### 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 76R10 Free convection
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##### References:
 [1] Adhikari, D.; Cao, C.; Shang, H.; Wu, J.; Xu, X.; Ye, Z., Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Differential Equations, 260, 1893-1917, (2016) · Zbl 1328.35161 [2] 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 [3] Ambrosetti, A.; Prodi, G., (A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, vol. 34, (1995), Cambridge Univ. Press Cambridge) · Zbl 0818.47059 [4] Bahouri, H.; Chemin, J.-Y.; Danchin, R., (Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, (2011), Springer) [5] Calderón, A. P., Commutators of singular integral operators, Proc. Natl. Acad. Sci. USA, 53, 1092-1099, (1965) · Zbl 0151.16901 [6] 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 [7] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084 [8] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, (1981), Dover Publications, Inc. · Zbl 0142.44103 [9] Chemin, J.-Y., Perfect incompressible fluids, (1998), Oxford University Press [10] Chen, Q.; Miao, C.; Zhang, Z., A new Bernstein inequality and the 2D dissipative quasigeostrophic equation, Comm. Math. Phys., 271, 821-838, (2007) · Zbl 1142.35069 [11] Coifman, R.; Meyer, Y., On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212, 315-331, (1975) · Zbl 0324.44005 [12] Constantin, P.; Vicol, V., Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22, 1289-1321, (2012) · Zbl 1256.35078 [13] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249, 3, 511-528, (2004) · Zbl 1309.76026 [14] 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 [15] 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 [16] Farhat, A.; Jolly, M.; Titi, E., Continuous data assimilation for the 2D bnard convection through velocity measurements alone, Physica D, 303, 59-66, (2015) · Zbl 1364.76053 [17] Feireisl, E.; Novotný, A., (Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, (2009), Birkhauser Verlag Basel) [18] Foias, C.; Manley, O.; Temam, R., Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal. TMA, 11, 8, 939-967, (1987) · Zbl 0646.76098 [19] Hmidi, T., On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal. PDE, 4, 247-284, (2011) · Zbl 1264.35173 [20] Hmidi, T.; Keraani, S., On the global well-posedness of the two-dimensional Boussinesq system with a zero viscosity, Indiana Univ. Math. J., 58, 4, 1591-1618, (2009) · Zbl 1178.35303 [21] 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 [22] 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 [23] Hmidi, T.; Zerguine, M., On the global well-posedness of the Euler-Boussinesq system with fractional dissipation, Physica D, 239, 15, 1387-1401, (2010) · Zbl 1194.35329 [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] 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 [26] Ju, N., The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255, 161-181, (2005) · Zbl 1088.37049 [27] 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 [28] Larios, A.; Lunasin, E.; Titi, E., 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 [29] Li, J.; Titi, E., Global well-posedness of strong solutions to a tropical climate model, Discrete Contin. Dyn. Syst., 36, 4495-4516, (2016) · Zbl 1339.35325 [30] Ma, T.; Wang, S., Rayleigh Bénard convection: dynamics and structure in the physical space, Commun. Math. Sci., 5, 3, 553-574, (2007) · Zbl 1133.35426 [31] Majda, A.; Bertozzi, A., 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] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York · Zbl 0713.76005 [34] Rabinowitz, P. H., Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch. Ration. Mech. Anal., 29, 32-57, (1968) · Zbl 0164.28704 [35] Stefanov, A.; Wu, J., A global regularity result for the 2D Boussinesq equations with critical dissipation, math.AP · Zbl 1420.35263 [36] Wu, G.; Xue, L., Global well-posedness for the 2D inviscid Bénard system with fractional diffusivity and yudovich’s type data, J. Differential Equations, 253, 100-125, (2012) · Zbl 1305.35119 [37] Wu, J.; Xu, X.; Xue, L.; Ye, Z., Regularity results for the 2D Boussinesq equations with critical and supercritical dissipation, Commun. Math. Sci., 14, 1963-1997, (2016) · Zbl 1358.35136 [38] 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 [39] Xu, X.; Xue, L., Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation, J. Differential Equations, 256, 3179-3207, (2014) · Zbl 1452.76030 [40] Xu, F.; Yuan, J., On the global well-posedness for the 2D Euler-Boussinesq system, Nonlinear Anal. RWA, 17, 137-146, (2014) · Zbl 1297.35193 [41] 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 [42] Ye, Z., Blow-up criterion of smooth solutions for the Boussinesq equations, Nonlinear Anal., 110, 97-103, (2014) · Zbl 1300.35109 [43] Ye, Z., On the regularity criterion for the 2D Boussinesq equations involving the temperature, Appl. Anal., 95, 615-626, (2016) · Zbl 1335.35207 [44] Ye, Z., A note on global regularity results for 2D Boussinesq equations with fractional dissipation, Ann. Polon. Math., 117, 3, 231-247, (2016) · Zbl 1358.35138 [45] 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 [46] Ye, Z.; Xu, X., Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, J. Differential Equations, 260, 6716-6744, (2016) · Zbl 1341.35135
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