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

Communications on Pure and Applied Mathematics

Volumn 60, Issue 9, 2007, Pages 1293-1318

  Wang, Xiaoming ^a,b  

^a FLORIDA STATE UNIVERSITY   (United States)

^b FUDAN UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34547829711     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20170     Document Type: Article

References (46)

1. Babin, A. V.; Vishik, M. I, Attractors of evolution equations. Studies in Mathematics and Its Applications, 25. North-Holland, Amsterdam, 1992.
2. Ball, J. M. Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. Mechanics .-from theory to computation, 447-474. Springer, New York, 2000.
3. Bodenschatz, E.; Pesch, W.; Ahlers, G. Recent developments in Rayleigh-Bénard convection. Annual review of fluid mechanics, Vol. 32, 709-778. Annual Review of Fluid Mechanics, 32. Annual Reviews, Palo Alto, Calif., 2000.
4. Busse, F. H. Fundamentals of thermal convection. Mantle convection: plate tectonics and global dynamics, 23-95. Montreux: Gordon and Breach, New York, 1989.
5. Busse, F. H,; Joseph, D. D. Bounds for heat transfer in a porous layer. J. Fluid Mech. 54 (1972), no. 3, 521-543.
6. Chandrasekhar, S. Hydrodynamic and hydromagnetic stability. The International Series of Monographs on Physics. Clarendon, Oxford, 1961.
7. Chen, Z.-M.; Price, W. G. Time dependent periodic Navier-Stokes flows on a two-dimensional torus. Comm. Math. Phys. 179 (1996), no. 3, 577-597.
8. Constantin, P.; Doering, C. R. Infinite Prandtl number convection. J. Statist. Phys. 94 (1999), no. 1-2, 159-172.
9. Constantin, P.; Foias, C. Navier-Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, 1988.
10. Constantin, P.; Hallstrom, C.; Poutkaradze, V. Logarithmic bounds for infinite Prandtl number rotating convection. J. Math. Phys. 42 (2001), no. 2, 773-783.
11. Doering, C. R.; Constantin, P. Bounds for heat transport in a porous layer. J. Fluid Mech. 376 (1998), 263-296.
12. Doering, C. R.; Constantin, P. On upper bounds for infinite Prandtl number convection with or without rotation. J. Math. Phys. 42 (2001), no. 2, 784-795.
13. Doering, C.R.; Gibbon, J. D. Applied analysis of the Navier-Stokes equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1995.
14. Doering, C. R.; Otto, F.; Reznikoff, M. G. Bounds on vertical heat transport for infinite Prandtl number Rayleigh-Bénard convection. J. Fluid Mech. 560 (2006), 229-241.
15. Doering, C. R.; Wang, X. Attractor dimension estimates for two-dimensional shear flows. Nonlinear waves and solitons in physical systems (Los Alamos, NM, 1997). Phys. D 123 (1998), no. 1-4, 206-222.
16. Doering, C. R.; Wang, X. Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number. Submitted.
17. Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R. Exponential attractors for dissipative evolution equations. RAM: Research in Applied Mathematics, 37. Masson, Paris; Wiley, Chichester, 1994.
18. Fabrie, P. Existence, unicité et comportement asymptotique de la solution d'un problème de convection en milieu poreux. C. R. Acad. Sci. Paris Sér. IMath. 295 (1982), no. 6, 423-425.
19. Fabrie, P.; Galusinski, C.; Miranville, A.; Zelik, S. Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications. Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 211-238.
20. Foias, C.; Manley, O.; Rosa, R.; Temam, R. Navier-Stokes equations and turbulence. Encyclopedia of Mathematics and Its Applications, 83. Cambridge University Press, Cambridge, 2001.
21. Gatti, S.; Grasselli, M.; Miranville, A.; Pata, V. A construction of a robust family of exponential attractors. Proc. Amer. Math. Soc. 134 (2006), no. 1, 117-127.
22. Getling, A. V. Rayleigh-Bénard convection. Structures and dynamics. Advanced Series in Nonlinear Dynamics, 11. World Scientific, River Edge, N.J., 1998.
23. Grossmann, S.; Lohse, D. Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407 (2000), 27-56.
24. Hale, J. K. Asymptotic behavior of dissipative systems. Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, R.I., 1988.
25. Judovič, V. I. An example of the loss of stability and the generation of a secondary flow of a ' fluid in a closed container. Mat. Sb. (N.S.) 74 (116) (1967), 565-579.
26. Kadanoff, L. P. Turbulent heat flow: structures and scaling. Physics Today 54 (2001), 34-39.
27. Ladyzhenskaya, O. Attractors for semigroups and evolution equations. Lezioni Lincee. Cambridge University Press, Cambridge, 1991.
28. Lax, P. D. Functional analysis. Pure and Applied Mathematics. Wiley, New York, 2002.
29. Ly, H. V.; Titi, E. S. Global Gevrey regularity for the Bénard convection in a porous medium with zero Darcy-Prandtl number. J. Nonlinear Sci. 9 (1999), no. 3, 333-362.
30. Ma, T.; Wang, S. Dynamic bifurcation and stability in the Rayleigh-Bénard convection. Commun. Math. Sci. 2 (2004), no. 2, 159-183.
31. Majda, A. Introduction to PDEs and waves for the atmosphere and ocean. Courant Lecture Notes in Mathematics, 9. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, R.I., 2003.
32. Majda, A.; Bertozzi, A. L. Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.
33. Miranville, A. Lower bound on the dimension of the attractor for the Bénard problem with free surfaces. Nonlinear Anal. 25 (1995), no. 11, 1079-1094.
34. Miranville, A.; Wang, X. Robust exponential attractors to the Boussinesq system for convection at large Prandtl number. In preparation.
35. Miranville, A.; Ziane, M. On the dimension of the attractor for the Bénard problem with free surfaces. Russian J. Math. Phys. 5 (1997), no. 4, 489-502.
36. Rabinowitz, P. H. Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal. 29 (1968), 32-57.
37. Robinson, J. C. Infinite-dimensional dynamical systems. An introduction to dissipative parabolic PDEs and the theory of global attractors. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.
38. Siggia, E. D. High Rayleigh number convection. Annual review of fluid mechanics, Vol. 26, 137-168. Annual Reviews, Palo Alto, Calif., 1994.
39. Temam, R. Navier-Stokes equations and nonlinear functional analysis. 2nd ed. CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1995.
40. Temam, R. Infinite-dimensional dynamical systems in mechanics and physics. 2nd ed. Applied Mathematical Sciences, 68. Springer, New York, 1997.
41. Temam, R. Navier-Stokes equations. Theory and numerical analysis. Reprint of the 1984 edition. AMS Chelsea, Providence, R.I., 2001.
42. Titi, E. S.; Wang, X. Small Darcy-Prandtl number convection in porous medium. In preparation.
43. Tritton, D. J. Physical fluid dynamics. 2nd ed. Oxford Science Publications. Clarendon, Oxford University Press, New York, 1988.
44. Wang, X. Infinite Prandtl number limit of Rayleigh-Bénard convection. Comm. Pure Appl. Math. 57 (2004), no. 10, 1265-1282.
45. Wang, X. Large Prandtl number behavior of the Boussinesq system of Rayleigh-Bénard convection. Appl. Math. Lett. 17 (2004), no. 7, 821-825.
46. Wang, X. 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.