×

zbMATH — the first resource for mathematics

Global regularity for the 2D Boussinesq equations with temperature-dependent viscosity. (English) Zbl 1433.35265
Summary: This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in [H. Abidi and P. Zhang, Adv. Math. 305, 1202–1249 (2017; Zbl 1353.35220)] and [X. Zhai et al., J. Differ. Equations 267, No. 1, 364–387 (2019; Zbl 1414.35153)]. In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient.
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
35R11 Fractional partial differential equations
42B25 Maximal functions, Littlewood-Paley theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abidi, H.; Zhang, P., On the global well-posedness of 2-D Boussinesq system with variable viscosity, Adv. Math., 305, 1202-1249 (2017) · Zbl 1353.35220
[2] Adhikari, D.; Cao, C.; Shang, H.; Wu, J.; Xu, X.; Ye, Z., Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Differ. Equ., 260, 1893-1917 (2016) · Zbl 1328.35161
[3] Bahouri,H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 343. Springer, Berlin (2011) · Zbl 1227.35004
[4] Cannon, J.; Dibenedetto, E., The Initial Value Problem for the Boussinesq Equation with Data in \(L^p\). Lecture Notes in Mathematics (1980), Berlin: Springer, Berlin · Zbl 0429.35059
[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] 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
[8] Chen, Q.; Miao, C.; Zhang, Z., A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation, Commun. Math. Phys., 271, 821-838 (2007) · Zbl 1142.35069
[9] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249, 511-528 (2004) · Zbl 1309.76026
[10] 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
[11] 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
[12] 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
[13] Hou, Ty; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12 (2005) · Zbl 1274.76185
[14] 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
[15] 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
[16] 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
[17] 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
[18] 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
[19] Lorca, S.; Boldrini, J., The initial value problem for a generalized Boussinesq model, Nonlinear Anal., 36, 457-480 (1999) · Zbl 0930.35136
[20] Majda, A.; Bertozzi, A., Vorticity and Incompressible Flow (2001), Cambridge: Cambridge University Press, Cambridge
[21] Miao, C.; Xue, L., On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differ. Equ. Appl., 18, 707-735 (2011) · Zbl 1235.76020
[22] Pedlosky, J., Geophysical Fluid Dynamics (1987), New York: Springer, New York · Zbl 0713.76005
[23] Schonbek, M., Large time behaviour of solutions to the Navier-Stokes equations, Commun. Partial Differ. Equ., 11, 733-763 (1986) · Zbl 0607.35071
[24] Silvestre, L., On the differentiablity of the solution to an equation with drift and fractional diffusion, Indiana Univ. Math. J., 61, 2, 557-584 (2012) · Zbl 1308.35042
[25] Stefanov, A.; Wu, J., A global regularity result for the 2D Boussinesq equations with critical dissipation, J. Anal. Math., 137, 269-290 (2019) · Zbl 1420.35263
[26] 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
[27] 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
[28] 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
[29] 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
[30] Xue, L.; Ye, Z., On the differentiability issue of the drift-diffusion equation with nonlocal L \(\rm \acute{e}\) vy-type diffusion, Pac. J. Math., 293, 471-4510 (2018) · Zbl 1379.35049
[31] 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
[32] Ye, Z., On the improved blow-up criterion for the 2D zero diffusivity Boussinesq equations with temperature-dependent viscosity, Appl. Anal., 97, 2037-2058 (2018) · Zbl 1398.35184
[33] Zhai, X.; Dong, B.; Chen, Z., Global well-posedness for 2-D Boussinesq system with the temperature-dependent viscosity and supercritical dissipation, J. Differ. Equ., 267, 364-387 (2019) · Zbl 1414.35153
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.