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February  2004, 10(1&2): 53-74. doi: 10.3934/dcds.2004.10.53

Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio

Björn Birnir 1, and  Nils Svanstedt 1,

1. 

University Of California, Santa Barbara, Ca 93106, United States, United States

Received  January 2002 Revised  May 2002 Published  October 2003

The Navier-Stokes equation driven by heat conduction is studied. It is proven that if the driving force is small then the solutions of the Navier-Stokes equation are ultimately regular. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl numer close to one, we prove the ultimate existence and regularity of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal $\mathcal B$-attractor. Examples of simple $\mathcal B$-attractors from pattern formation are given and a method to study their instabilities proposed.
Keywords: Rayleigh-Bénard problem, Existence theory.
Mathematics Subject Classification: 35Q30.
Citation: Björn Birnir, Nils Svanstedt. Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 53-74. doi: 10.3934/dcds.2004.10.53
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