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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

 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.
Citation: O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449
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