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

Björn Birnir; Nils Svanstedt

doi:10.3934/dcds.2004.10.53

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2004, Volume 10, Issue 1&2: 53-74. Doi: 10.3934/dcds.2004.10.53

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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

Received: December 31, 2001

Revised: April 30, 2002

Published: December 2003

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Abstract

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:

Existence theory,
Rayleigh-Bénard problem.

Mathematics Subject Classification: 35Q30.

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