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.
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Ball, J.: Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations. In: Mechanics: From Theory to Computation, pp. 447–474. Springer, New York (2000)
Birnir, B., Svanstedt, N.: Existence theory and strong attractors for the Rayleigh–Bénard problem with a large aspect ratio. Discrete Contin. Dyn. Syst. 10, 53–74 (2004)
Bondarevski, V.G.: Energetic systems and global attractors for the 3D Navier–Stokes equations. Nonlinear Anal. 30, 799–810 (1997)
Chepyzhov, V.V., Vishik, M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76, 913–964 (1997)
Foias, C., Temam, R.: The connection between the Navier–Stokes equations, dynamical systems, and turbulence theory. In: Directions in Partial Differential Equations, pp. 55–73. Academic Press (1987)
Foias, C., Manley, O.P., Rosa, R.R., Temam, R.: Navier–Stokes Equations and Turbulence. Cambridge University Press, Cambridge (2001)
Ladyzhenskaya, O.A.: Attractors of Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)
Lions, J.L.: Quelques Méthodes de Résolutions des Problèmes Aux Limites Non Linéaires. Dunod, Gauthier-Villars, Paris (1969)
Kapustyan, A.V., Melnik, V.S., Valero, J.: Attractors of multivalued dynamical processes generated by phase-field equations. Int. J. Bifurc. Chaos 13, 1969–1984 (2003)
Kapustyan, O.V.: On existence of minimal weakly attracting set in the phase space for the 3-D Navier–Stokes system. Dopov. Nac. Akad. Nauk Ukr. 12, 18–22 (2005)
Kapustyan, O.V., Melnik, V.S., Valero, J., Yasinsky, V.V.: Global Attractors of Multivalued Dynamical Systems and Evolution Equations Without Uniqueness. Naukova Dumka, Kyiv (2008)
Kapustyan, O.V., Melnik, V.S., Valero, J.: A weak attractors and properties of solutions for the three-dimensional Bénard problem. Discrete Contin. Dyn. Syst. 18, 449–481 (2007)
Kapustyan, O.V., Valero, J.: Weak and strong attractors for the 3D Navier–Stokes system. J. Differ. Equ. 240, 249–278 (2007)
Melnik, V.S., Valero, J.: On attractors of multi-valued semi-flows and differential inclusions. Set-Valued Anal. 6, 83–111 (1998)
Norman, D.E.: Chemically reacting fluid flows: weak solutions and global attractors. J. Differ. Equ. 152, 75–135 (1999)
Robinson, J.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)
Sell, G.: Global attractors for the three-dimensional Navier–Stokes equations. J. Dyn. Differ. Equ. 8, 1–33 (1996)
Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New-York (2002)
Simsen, J., Gentile, C.: On attractors for multivalued semigroups defined by generalized semiflows. Set-Valued Anal. 16, 105–124 (2008)
Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1979)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York (1988)
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Kapustyan, A.V., Pankov, A.V. & Valero, J. On Global Attractors of Multivalued Semiflows Generated by the 3D Bénard System. Set-Valued Var. Anal 20, 445–465 (2012). https://doi.org/10.1007/s11228-011-0197-5
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DOI: https://doi.org/10.1007/s11228-011-0197-5
Keywords
- Three-dimensional Boussinesq equations
- Bénard problem
- Three-dimensional Navier–Stokes equations
- Set-valued dynamical system
- Global attractor