Shi, Jian-guo; Wang, Shu; Wang, Ke; He, Feng-ying The perturbed problem on the Boussinesq system of Rayleigh-Bénard convection. (English) Zbl 1300.35103 Acta Math. Appl. Sin., Engl. Ser. 30, No. 1, 75-88 (2014). Summary: In this paper, the infinite Prandtl number limit of Rayleigh-Bénard convection is studied. For well prepared initial data, the convergence of solutions in \(L^\infty(0,t;H^2(G))\) is rigorously justified by analysis of asymptotic expansions. MSC: 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs 35K57 Reaction-diffusion equations 76R05 Forced convection Keywords:infinite Prandtl number limit; Rayleigh-Bénard convection; Boussinesq system; asymptotic; expansions; iterative techniques PDFBibTeX XMLCite \textit{J.-g. Shi} et al., Acta Math. Appl. Sin., Engl. Ser. 30, No. 1, 75--88 (2014; Zbl 1300.35103) Full Text: DOI References: [1] Bodenschatz, E., Pesch, W., Ahlers, G. Recent developments in Rayleigh-Bénard convection. Annual review of fluid mechanics, 32: 709–778 (2000) · Zbl 0988.76033 · doi:10.1146/annurev.fluid.32.1.709 [2] Busse, F.H. Fundamentals of thermal convection. Mantle convection: plate tectonics and fluid dynamics, 23–95. W.R. Peltier, ed. The Fluid Mechanics of Astrophysics and Geophysics, Vol. 4, Gordon and Breach, New York, 1989 [3] Chandrasekhar, S. Hydrodynamic and hydromagnetic stability. The International series of Monographs on physics. Clarendon, Oxford, 1961 · Zbl 0142.44103 [4] Getling, A.V. Rayleigh-Bénard convection: structures and dynamics. Advanced Series in Nonlinear Dynamics, 11. World Scientific Publishing Co., Inc., River Edge, NJ, 1998 · Zbl 0910.76001 [5] Ladyzhenskaya, O.A. The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, London, New York, 1969 · Zbl 0184.52603 [6] Lions, J.L. Quelques méthods de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969 [7] Lions, J.L. Perturbations singuliéres dans les problémes aux limites et en controle optimal. Lecture notes in mathematics, 323, Springer-verlag, Berlin, New York, 1973 [8] Shi, J.G., Wang, K., Wang, S. The initial layer problem and infinite Prandtl number limit of Rayleigh-Bénard convection. Commun. Math. Sci., 5(1): 53–66 (2007) · Zbl 1157.35311 · doi:10.4310/CMS.2007.v5.n1.a2 [9] Shi, J.G., Zhou, H.Y. Existence and regularity of the solution to the Stokes coupling system. Journal of Lanzhou University (Natural Sciences), 44(5): 103–107 (2008) [10] Siggia, E.D. High Rayleigh number convection. Annual review of fluid mechanics, 26: 137–168 (1994) · Zbl 0800.76425 · doi:10.1146/annurev.fl.26.010194.001033 [11] Simon, J. Compact sets in the space L p(0, T;B). Ann. Mat. Pura Appl., 146(4): 65–96 (1987) · Zbl 0629.46031 · doi:10.1007/BF01762360 [12] Tritton, D.J. Physical fluid dynamics. 2nd ed. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1988 · Zbl 0383.76001 [13] Wang, X.M. Infinite Prandtl number limit of Rayleigh-Bénard convection. Communications on Pure and Applied Mathematics, 57(10): 1265–1282 (2004) · Zbl 1112.76032 · doi:10.1002/cpa.3047 [14] Wang, X.M. A note on long time behavior of solutions to the Boussinesq system atlarge Prandtl number. Nonlinear partial differential equations and related analysis, 315–323, Contemp. Math., 371, Amer. Math. Soc., Providence, RI, 2005 · Zbl 1080.35095 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.