Set-Valued and Variational Analysis

, Volume 20, Issue 3, pp 445–465

On Global Attractors of Multivalued Semiflows Generated by the 3D Bénard System

  • Alexey V. Kapustyan
  • Alexey V. Pankov
  • J. Valero
Article

DOI: 10.1007/s11228-011-0197-5

Cite this article as:
Kapustyan, A.V., Pankov, A.V. & Valero, J. Set-Valued Var. Anal (2012) 20: 445. doi:10.1007/s11228-011-0197-5

Abstract

In this paper we prove the existence of solutions for the 3D Bénard system in the class of functions which are strongly continuous with respect to the second component of the vector (that is, the one corresponding to the parabolic equation). We construct then a multivalued semiflow generated by such solutions and obtain the existence of a global φ −attractor for the weak-strong topology. Moreover, a family of multivalued semiflows is defined on suitable convex bounded subsets of the phase space, proving for them the existence of a global attractor (which is the same for every semiflow of the family) for the weak-strong topology.

Keywords

Three-dimensional Boussinesq equations Bénard problem Three-dimensional Navier–Stokes equations Set-valued dynamical system Global attractor 

Mathematics Subject Classifications (2010)

35B40 35B41 35K55 35Q30 35Q35 37B25 58C06 

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Alexey V. Kapustyan
    • 1
  • Alexey V. Pankov
    • 2
  • J. Valero
    • 3
  1. 1.Institute for Applied System Analysis NASU, NADOS LaboratoryTaras Shevchenko National University of KyivKyivUkraine
  2. 2.Taras Shevchenko National University of KyivKyivUkraine
  3. 3.Centro de Investigación OperativaUniversidad Miguel Hernandez de ElcheElche (Alicante)Spain

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