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On the improved blow-up criterion for the 2D zero diffusivity Boussinesq equations with temperature-dependent viscosity. (English) Zbl 1398.35184
Summary: This paper examines the two-dimensional zero diffusivity Boussinesq equations with temperature-dependent viscosity in the whole space. Whether or not its classical solutions global in time existence is a difficult problem and remains open. The main goal of this paper is to establish several blow-up criteria. More precisely, it is proved that $$0<T<\infty$$ is the maximal time interval if and only if the viscosity $$\nu(\theta)$$ satisfies $\int^T_0 \| \nabla\nu(\theta) (t)\|^2_{\mathrm{BMO}}\mathrm{d}t=\infty,$ or the vorticity $$\omega$$ satisfies $\int^T_0 \|\omega (t)\|^2_{\mathrm{BMO}}\mathrm{d}t=\infty.$ As a direct application of the above results, several other blow-up criteria in the Besov spaces and the multiplier spaces are also established.

##### 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 35B44 Blow-up in context of PDEs
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##### References:
 [1] Majda, A; Bertozzi, A, Vorticity and incompressible flow, (2001), Cambridge University Press, Cambridge [2] Pedlosky, J, Geophysical fluid dynamics, (1987), Springer-Verlag, New York (NY) [3] Chae, D, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv Math, 203, 497-513, (2006) · Zbl 1100.35084 [4] Hou, TY; Li, C, Global well-posedness of the viscous Boussinesq equations, Discrete Contin Dyn Syst, 12, 1-12, (2005) · Zbl 1274.76185 [5] Hmidi, T; Keraani, S, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv Differ Equ, 12, 461-480, (2007) · Zbl 1154.35073 [6] 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 [7] Lai, M; Pan, R; Zhao, K, Initial boundary value problem for 2D viscous Boussinesq equations, Arch Ration Mech Anal, 199, 739-760, (2011) · Zbl 1231.35171 [8] Constantin, P; Vicol, V, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom Funct Anal, 22, 1289-1321, (2012) · Zbl 1256.35078 [9] Hmidi, T; Keraani, S; Rousset, F, Global well-posedness for a Boussinesq–Navier–Stokes system with critical dissipation, J Differ Equ, 249, 2147-2174, (2010) · Zbl 1200.35228 [10] Hmidi, T; Keraani, S; Rousset, F, Global well-posedness for Euler–Boussinesq system with critical dissipation, Commun Partial Differ Equ, 36, 420-445, (2011) · Zbl 1284.76089 [11] 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 [12] Miao, C; Xue, L, On the global well-posedness of a class of Boussinesq–Navier–Stokes systems, Nonlinear Differ Equ Appl, 18, 707-735, (2011) · Zbl 1235.76020 [13] Stefanov, A; Wu, J, A global regularity result for the 2D Boussinesq equations with critical dissipation, J Anal Math · Zbl 1420.35263 [14] Wu, J; Xu, X; Xue, L, Regularity results for the 2D Boussinesq equations with critical and supercritical dissipation, Commun Math Sci, 14, 1963-1997, (2016) · Zbl 1358.35136 [15] Ye, Z, Global smooth solution to the 2D Boussinesq equations with fractional dissipation, Math Methods Appl Sci · Zbl 1369.35072 [16] Ye, Z; Xu, X, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, J Differ Equ, 260, 6716-6744, (2016) · Zbl 1341.35135 [17] Cao, C; Wu, J, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch Rational Mech, 208, 985-1004, (2013) · Zbl 1284.35140 [18] 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 [19] Adhikari, D; Cao, C; Shang, H, Global regularity results for the 2D Boussinesq equations with partial dissipation, J Differ Equ, 260, 1893-1917, (2016) · Zbl 1328.35161 [20] Wu, G; Zheng, X, Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation, J Differ Equ, 255, 2891-2926, (2013) · Zbl 1360.76076 [21] 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 [22] Li, D; Xu, X, Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity, Dyn PDE, 10, 255-265, (2013) · Zbl 1302.35319 [23] He, M, On the blowup criteria and global regularity for the non-diffusive Boussinesq equations with temperature-dependent viscosity coefficient, Nonlinear Anal, 144, 93-109, (2016) · Zbl 1350.35154 [24] Huang, A, The global well-posedness and global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity, Nonlinear Anal, 113, 401-429, (2015) · Zbl 1304.35491 [25] Jiu, Q; Liu, J, Global-wellposedness of 2D Boussinesq equations with mixed partial temperature-dependent viscosity and thermal diffusivity, Nonlinear Anal, 132, 227-239, (2016) · Zbl 1335.35197 [26] Li, H; Pan, R; Zhang, W, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion, J Hyperbolic Differ Equ, 12, 469-488, (2015) · Zbl 1328.35173 [27] Sun, Y; Zhang, Z, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J Differ Equ, 255, 1069-1085, (2013) · Zbl 1284.35322 [28] Cannon, J; DiBenedetto, E, The initial value problem for the Boussinesq equation with data in Lp, 771, 129-144, (1980), Springer, Berlin [29] Lorca, S; Boldrini, J, The initial value problem for a generalized Boussinesq model, Nonlinear Anal, 36, 457-480, (1999) · Zbl 0930.35136 [30] Lorca, S; Boldrini, J, The initial value problem for a generalized Boussinesq model: regularity and global existence of strong solutions, Mat Contemp, 11, 71-94, (1996) · Zbl 0861.35080 [31] Chen, Q; Jiang, L, Global well-posedness for the 2-D Boussinesq system with temperature-dependent thermal diffusivity, Colloq Math, 135, 187-199, (2014) · Zbl 1307.35214 [32] Bahouri, H; Chemin, JY; Danchin, R, Fourier analysis and nonlinear partial differential equations, 343, (2011), Springer, Heidelberg · Zbl 1227.35004 [33] Lemarié-Rieusset, PG, Recent developments in the Navier-Stokes problem, (2002), Chapman & Hall/CRC, London · Zbl 1034.35093 [34] Zheng, X, A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component, J Differ Equ, 256, 283-309, (2014) · Zbl 1331.35266 [35] 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 [36] Brezis, H; Gallouet, T, Nonlinear schrodinger evolution equations, Nonlinear Anal, 4, 677-681, (1980) · Zbl 0451.35023 [37] Kato, T; Ponce, G, Commutator estimates and the Euler and Navier-Stokes equations, Commun Pure Appl Math, 41, 891-907, (1988) · Zbl 0671.35066 [38] Ogawa, T, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J Math Anal, 34, 1318-1330, (2003) · Zbl 1036.35082
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