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Small global solutions to the damped two-dimensional Boussinesq equations. (English) Zbl 1290.35193
The paper considers the damped two-dimensional Boussinesq equations. By positioning the solutions in some homogeneous Besov space, it is shown that there exists a unique global solution to the damped Boussinesq equations under a small assumption of the given initial data in the same homogeneous Besov space.
Reviewer: Cheng He (Beijing)

35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
Full Text: DOI arXiv
[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] Bahouri, H.; Chemin, J.-Y.; Danchin, R., Fourier analysis and nonlinear partial differential equations, (2011), Springer · Zbl 1227.35004
[4] Bergh, J.; Löfström, J., Interpolation spaces, an introduction, (1976), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0344.46071
[5] Brandolese, L.; Schonbek, M., Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Amer. Math. Soc., 364, 5057-5090, (2012) · Zbl 1368.35217
[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., Remarks on the blow-up of the Euler equations and the related equations, Comm. Math. Phys., 245, 539-550, (2004) · Zbl 1145.35443
[8] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084
[9] Chae, D.; Wu, J., The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230, 1618-1645, (2012) · Zbl 1248.35156
[10] Constantin, P.; Doering, C. R., Infinite Prandtl number convection, J. Stat. Phys., 94, 159-172, (1999) · Zbl 0935.76083
[11] Constantin, P.; Vicol, V., Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22, 1289-1321, (2012) · Zbl 1256.35078
[12] 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
[13] 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
[14] 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
[15] E, W.; Shu, C., Small-scale structures in Boussinesq convection, Phys. Fluids, 6, 49-58, (1994) · Zbl 0822.76087
[16] Gill, A. E., Atmosphere-Ocean dynamics, (1982), Academic Press London
[17] Han, P.; Schonbek, M., Large time decay properties of solutions to a viscous Boussinesq system in a half space, Adv. Differential Equations, 19, 1/2, 87-132, (2014) · Zbl 1286.35206
[18] Hmidi, T., On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal. PDE, 4, 247-284, (2011) · Zbl 1264.35173
[19] 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
[20] 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
[21] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185
[22] D. KC, D. Regmi, L. Tao, J. Wu, Generalized 2D Euler-Boussinesq equations with a singular velocity, preprint. · Zbl 1291.35221
[23] Lai, M.-J.; 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
[24] 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
[25] 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
[26] Majda, A. J.; Bertozzi, A. L., Vorticity and incompressible flow, (2001), Cambridge University Press
[27] 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
[28] Miao, C.; Wu, J.; Zhang, Z., Littlewood-Paley theory and its applications in partial differential equations of fluid dynamics, (2012), Science Press Beijing, China, (in Chinese)
[29] Moffatt, H. K., Some remarks on topological fluid mechanics, (Ricca, R. L., An Introduction to the Geometry and Topology of Fluid Flows, (2001), Kluwer Academic Publishers Dordrecht, The Netherlands), 3-10 · Zbl 1100.76500
[30] 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, Surikaisekikenkyusho Kokyuroku, vol. 1234, (2001)), 127-145
[31] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York · Zbl 0713.76005
[32] Runst, T.; Sickel, W., Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations, (1996), Walter de Gruyter Berlin, New York · Zbl 0873.35001
[33] Triebel, H., Theory of function spaces II, (1992), Birkhäuser Verlag · Zbl 0763.46025
[34] 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
[35] Xu, X., Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Anal., 72, 677-681, (2010) · Zbl 1177.76024
[36] Xu, X.; Ye, Z., The lifespan of solutions to the inviscid 3D Boussinesq system, Appl. Math. Lett., 26, 854-859, (2013) · Zbl 1314.35113
[37] Zhao, K., 2D inviscid heat conductive Boussinesq equations on a bounded domain, Michigan Math. J., 59, 329-352, (2010) · Zbl 1205.35048
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