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Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number. (English) Zbl 1143.35351
Summary: This is the third work in a series of our study of Rayleigh-Bénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for Rayleigh-Bénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form $\text{Ra}^{1/3}(\ln \text{Ra})^{1/3} + c \frac {\text{Ra}^{7/2}\ln \text{Ra}}{\text{Pr}^2}$ which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76E06 Convection in hydrodynamic stability 76R05 Forced convection
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 [1] Amati, Phys Fluids 17 pp 121701-1– (2005) [2] Weak convergence of measures: Applications in probability. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, 5. Society for Industrial and Applied Mathematics, Philadelphia, 1971. · Zbl 0271.60009 · doi:10.1137/1.9781611970623 [3] ; ; Recent developments in Rayleigh-Bénard convection. Annual review of fluid mechanics, vol. 32, 709–778. Annual Reviews, Palo Alto, Calif., 2000. [4] Fundamentals of thermal convection. Mantle convection: Plate tectonics and global dynamics, 23–95. Gordon and Breach, New York, 1989. [5] Chae, J Math Anal Appl 155 pp 437– (1991) [6] Chae, J Math Anal Appl 155 pp 460– (1991) [7] Hydrodynamic and hydro-magnetic stability. International Series of Monographs on Physics. Clarendon, Oxford, 1961. [8] Constantin, Nonlinearity 9 pp 1049– (1996) [9] Constantin, J Statist Phys 94 pp 159– (1999) [10] ; Navier-Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, 1988. [11] Constantin, J Math Phys 42 pp 773– (2001) [12] Constantin, J Math Phys 38 pp 3031– (1997) [13] ; Ergodicity for infinite dimensional systems. Cambridge University Press, Cambridge–New York, 1996. · Zbl 0849.60052 · doi:10.1017/CBO9780511662829 [14] Doering, J Math Phys 42 pp 784– (2001) [15] ; Applied analysis of the Navier-Stokes equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1995. · Zbl 0838.76016 · doi:10.1017/CBO9780511608803 [16] Doering, J Fluid Mech 560 pp 229– (2006) [17] Stochastic hydrodynamics. Current developments in mathematics, 2000, 109–147. International, Somerville, Mass., 2001. [18] ; ; ; Navier-Stokes equations and turbulence. Encyclopedia of Mathematics and Its Applications, 83. Cambridge University Press, Cambridge, 2001. · doi:10.1017/CBO9780511546754 [19] Rayleigh-Bénard convection. Structures and dynamics. Advanced Series in Nonlinear Dynamics, 11. World Scientific, River Edge, N.J., 1998. · doi:10.1142/3097 [20] Grossmann, J Fluid Mech 407 pp 27– (2000) [21] Howard, J Fluid Mech 17 pp 405– (1963) [22] Ierley, J Fluid Mech 560 pp 159– (2006) [23] Kadanoff, Phys Today 54 pp 34– (2001) [24] Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience, New York, 2002. [25] Lee, J Statist Phys 117 pp 929– (2004) [26] Ma, Commun Math Sci 2 pp 159– (2004) · Zbl 1133.76315 · doi:10.4310/CMS.2004.v2.n2.a2 [27] ; Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. · Zbl 0983.76001 [28] ; Non-linear dynamics and statistical theories for basic geophysical flows. Cambridge University Press, Cambridge, 2006. · doi:10.1017/CBO9780511616778 [29] ; Statistical fluid mechanics: Mechanics of turbulence. MIT Press, Cambridge, Mass., 1975. [30] Nikolaenko, J Fluid Mech 523 pp 251– (2005) [31] Rabinowitz, Arch Rational Mech Anal 29 pp 32– (1968) [32] High Rayleigh number convection. Annual review of fluid mechanics, vol. 26, 137–168. Annual Reviews, Palo Alto, Calif., 1994. [33] Infinite-dimensional dynamical systems in mechanics and physics. 2nd ed. Applied Mathematical Sciences, 68. Springer, New York, 1997. · doi:10.1007/978-1-4612-0645-3 [34] Navier-Stokes equations. Theory and numerical analysis. Reprint of the 1984 edition. AMS Chelsea, Providence, R.I., 2001. [35] ; Mathematical problems of statistical hydromechanics. Mathe-matics and Its Applications. Kluwer, Dordrecht, The Netherlands, 1988. · Zbl 0688.35077 · doi:10.1007/978-94-009-1423-0 [36] An introduction to ergodic theory. Graduate Texts in Mathematics, 79. Springer, New York–Berlin, 2000. · Zbl 0958.28011 [37] Wang, Comm Pure Appl Math 57 pp 1265– (2004) [38] Wang, Appl Math Lett 17 pp 821– (2004) [39] A note on long time behavior of solutions to the Boussinesq system at large Prandtl number. Nonlinear partial differential equations and related analysis, 315–323. Contemporary Mathematics, 371. American Mathematical Society, Providence, R.I., 2005. · Zbl 1080.35095 · doi:10.1090/conm/371/06862 [40] Wang, Comm Pure Appl Math 60 pp 1293– (2007) [41] Wang, Phys D
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