Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 53 - 74, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.53

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Björn Birnir - University Of California, Santa Barbara, Ca 93106, United States (email)
Nils Svanstedt - University Of California, Santa Barbara, Ca 93106, United States (email)

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

Received: January 2002;      Revised: May 2002;      Available Online: October 2003.