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February  2004, 10(1&2): 89-116. doi: 10.3934/dcds.2004.10.89

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

Marco Cabral 1, , Ricardo Rosa 1, and  Roger Temam 2,

1. 

Department of Applied Mathematics, Federal University of Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ 21945–970, Brazil, Brazil
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  October 2003

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.
Keywords: noncompact systems, fractal dimension, Bénard problem, global attractor, infinite dimensional dynamical system..
Mathematics Subject Classification: 35Q35, 76D03, 37L3.
Citation: Marco Cabral, Ricardo Rosa, Roger Temam. Existence and dimension of the attractor for the Bénard problem on channel-like domains. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 89-116. doi: 10.3934/dcds.2004.10.89
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