Existence and dimension of the attractor for the Bénard problem on channel-like domains

Marco Cabral; Ricardo Rosa; Roger Temam

doi:10.3934/dcds.2004.10.89

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2004, Volume 10, Issue 1&2: 89-116. Doi: 10.3934/dcds.2004.10.89

This issue Previous Article On quasi-periodic lattice Schrödinger operators Next Article On the fractal dimension of invariant sets: Applications to Navier-Stokes equations

Existence and dimension of the attractor for the Bénard problem on channel-like domains

1.
Department of Applied Mathematics, Federal University of Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ 21945–970
2.
The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405

Received: November 2002

Revised: May 2003

Published: January 2004

Abstract / Introduction Related Papers Cited by

Abstract

The Bénard problem, a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the temperature is considered. Non-homogeneous boundary conditions, external force and heat source in dual function spaces, and an arbitrary spatial domain (possibly nonsmooth and unbounded) as long as the Poincaré inequality holds on it (channel-like domain) are allowed. Moreover our approach, unlike in previous works, avoids the use of the maximum principle which would be problematic in this context. The mathematical formulation of the problem, the existence of global solution and the existence and finite dimensionality of the global attractor are proved.

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Mathematics Subject Classification: 35Q35, 76D03, 37L30.

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Existence and dimension of the attractor for the Bénard problem on channel-like domains

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