Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number

We study asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh-Benard convection at large Prandtl number. In particular, we show that the global attractors to the Boussinesq system for Rayleigh-Benard convection converge to that of the infinite-Prandtl-number model for convection as the Prandtl number approaches infinity. This offers partial justification of the infinite-Prandtl-number model for convection as a valid simplified model for convection at large Prandtl number even in the long-time regime. c 2006 Wiley Periodicals, Inc.

[1]  P. Bergé,et al.  Rayleigh-bénard convection , 1984 .

[2]  Akira Ogawa,et al.  Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics , 2002 .

[3]  Charles R. Doering,et al.  Infinite Prandtl Number Convection , 1999 .

[4]  O. Ladyzhenskaya,et al.  Attractors for Semigroups and Evolution Equations , 1991 .

[5]  R. Temam.  Navier-Stokes Equations , 1977 .

[6]  Leo P. Kadanoff,et al.  Turbulent heat flow: Structures and scaling , 2001 .

[7]  Logarithmic Bounds for Infinite Prandtl Number Rotating Convection , 2001 .

[8]  Mahendra K. Verma,et al.  Recent developments in Rayleigh Bnard Convection , 2006 .

[9]  W. Price,et al.  Time dependent periodic Navier-Stokes flows on a two-dimensional torus , 1996 .

[10]  Roger Temam,et al.  Navier-Stokes Equations and Turbulence by C. Foias , 2001 .

[11]  P. Constantin,et al.  Bounds for heat transport in a porous layer , 1998, Journal of Fluid Mechanics.

[12]  Charles R. Doering,et al.  On upper bounds for infinite Prandtl number convection with or without rotation , 2001 .

[13]  Xiaoming Wang,et al.  Infinite Prandtl number limit of Rayleigh‐Bénard convection , 2004 .

[14]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[15]  Shouhong Wang,et al.  Dynamic Bifurcation and Stability in the Rayleigh-Benard Convection , 2004 .

[16]  A. Majda.  Introduction to PDEs and Waves in Atmosphere and Ocean , 2003 .

[17]  E. Titi,et al.  Global Gevrey Regularity for the Bénard Convection in a Porous Medium with Zero Darcy-Prandtl Number , 1999 .

[18]  Eberhard Bodenschatz,et al.  Recent Developments in Rayleigh-Bénard Convection , 2000 .

[19]  A. Miranville,et al.  On the dimension of the attractor for the B'enard problem with free surfaces , 1997 .

[20]  John M. Ball,et al.  Erratum to: Continuity Properties and Global Attractors of Generalized Semiflows and the Navier-Stokes Equations , 1997 .

[21]  A. Majda,et al.  Vorticity and incompressible flow , 2001 .

[22]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[23]  C. Doering,et al.  Applied analysis of the Navier-Stokes equations: Index , 1995 .

[24]  Alain Miranville,et al.  A construction of a robust family of exponential attractors , 2005 .

[25]  Xiaoming Wang.  A Note on Long Time Behavior of Solutions to the Boussinesq System at Large , 2004 .

[26]  M. Vishik,et al.  Attractors of Evolution Equations , 1992 .

[27]  James C. Robinson.  Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors , 2001 .

[28]  L. E. Fraenkel,et al.  NAVIER-STOKES EQUATIONS (Chicago Lectures in Mathematics) , 1990 .

[29]  Meinhard E. Mayer,et al.  Navier-Stokes Equations and Turbulence , 2008 .

[30]  S. Chandrasekhar.  Hydrodynamic and Hydromagnetic Stability , 1961 .

[31]  Attractor dimension estimates for two-dimensional shear flows , 1998 .

[32]  Edward A. Spiegel,et al.  Rayleigh‐Bénard Convection: Structures and Dynamics , 1998 .

[33]  Xiaoming Wang.  Stationary statistical properties of Rayleigh‐Bénard convection at large Prandtl number , 2008 .

[34]  W. Peltier.  Mantle convection : plate tectonics and global dynamics , 1989 .

[35]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[36]  Richard E. Mortensen,et al.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam) , 1991, SIAM Rev..

[37]  Detlef Lohse,et al.  Scaling in thermal convection: a unifying theory , 2000, Journal of Fluid Mechanics.

[38]  Charles R. Doering,et al.  Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh–Bénard convection , 2005, Journal of Fluid Mechanics.

[39]  Roger Temam,et al.  Navier–Stokes Equations and Nonlinear Functional Analysis: Second Edition , 1995 .

[40]  Xiaoming Wang,et al.  Large Prandtl number behavior of the Boussinesq system of Rayleigh-Bénard convection , 2004, Appl. Math. Lett..

[41]  A. Eden,et al.  Exponential Attractors for Dissipative Evolution Equations , 1995 .

[42]  Charles R. Doering,et al.  Applied analysis of the Navier-Stokes equations: Index , 1995 .

[43]  P. Rabinowitz,et al.  Existence and nonuniqueness of rectangular solutions of the Bénard problem , 1968 .

[44]  Sergey Zelik,et al.  Uniform exponential attractors for a singularly perturbed damped wave equation , 2003 .

[45]  J. M. Ball,et al.  Continuity Properties and Global Attractors of Generalized Semiflows and the Navier-Stokes Equations , 1997 .

[46]  J. Hart,et al.  High Rayleigh number β-convection , 1993 .

[47]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[48]  Lower bound on the dimension of the attractor for the Be´nard problem with free surfaces , 1995 .