# American Institute of Mathematical Sciences

January & 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

 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.
Citation: Björn Birnir, Nils Svanstedt. Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 53-74. doi: 10.3934/dcds.2004.10.53
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