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March  2007, 18(2&3): 449-481. doi: 10.3934/dcds.2007.18.449

A weak attractor and properties of solutions for the three-dimensional Bénard problem

O. V. Kapustyan 1, , V. S. Melnik 2, and  José Valero 3,

1. 

Department of Mathematics and Mechanics, Kiev Taras Shevchenko University, Volodymyrska str., 01033, Kiev, Ukraine
2. 

Institute of Applied System Analysis, Pr. Pobedy 37, 252056, Kiev, Ukraine
3. 

Centro de Investigación Operativa, Universidad Miguel Hernández, Avda Universidad s/n, 03202 Elche, Alicante, Spain

Received  March 2006 Revised  August 2006 Published  March 2007

In this paper we study the asymptotic behaviour of weak solutions for the three-dimensional Boussinesq equations (also known as the Bénard problem). First, we prove some regularity properties of the weak solutions of the system. Then we construct a one parameter familiy of multivalued semiflows and for them obtain the existence of a global attractor with respect to the weak topology of the phase space. Finally, we obtain a conditional result (valid only under an unproved hypothesis) stating the existence of a global attractor with respect to the strong topology.
Keywords: Bénard problem, global attractor., Three-dimensional Boussinesq equations, setvalued dynamical system.
Mathematics Subject Classification: 35B40, 35B41, 35K55, 35Q30, 37B25, 58C0.
Citation: O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449
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