Set-Valued and Variational Analysis

, Volume 21, Issue 3, pp 517–540 | Cite as

The Envelope Attractor of Non-strict Multivalued Dynamical Systems with Application to the 3D Navier–Stokes and Reaction–Diffusion Equations



Multivalued semiflows generated by evolution equations without uniqueness sometimes satisfy a semigroup set inclusion rather than equality because, for example, the concatentation of solutions satisfying an energy inequality almost everywhere may not satisfy the energy inequality at the joining time. Such multivalued semiflows are said to be non-strict and their attractors need only be negatively semi-invariant. In this paper the problem of enveloping a non-strict multivalued dynamical system in a strict one is analyzed and their attactors are compared. Two constructions are proposed. In the first, the attainability set mapping is extending successively to be strict at the dyadic numbers, which essentially means (in the case of the Navier–Stokes system) that the energy inequality is satisfied piecewise on successively finer dyadic subintervals. The other deals directly with trajectories and their concatenations, which are then used to define a strict multivalued dynamical system. The first is shown to be applicable to the three-dimensional Navier–Stokes equations and the second to a reaction–diffusion problem without unique solutions.


Multivalued dynamical systems Non-strict multivalued semiflows Non-strict and strict global attractors 3D Navier–Stokes equations Reaction–diffusion equations 

Mathematics Subject Classifications (2010)

35B40 35B41 35K55 35K57 35Q30 37B25 58C06 


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  1. 1.
    Anguiano, M., Caraballo, T., Real, J., Valero, J.: Pullback attractors for reaction–diffusion equations in some unbounded domains with an H  − 1-valued non-autonomous forcing term and without uniqueness of solutions. Discrete Contin. Dyn. Syst. Ser. B 14, 307–326 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Balibrea, F., Caraballo, T., Kloeden, P.E., Valero, J.: Recent developments in dynamical systems: three perspectives. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20, 2591–1636 (2010)Google Scholar
  3. 3.
    Ball, J.M.: 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)CrossRefGoogle Scholar
  4. 4.
    Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. American Mathematical Society, Providence, RI (2002)MATHGoogle Scholar
  5. 5.
    Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Structure and regularity of the global attractor of a reaction–diffusion equation with non-smooth nonlinear term. Preprint (2012). arXiv:1209.2010v1
  6. 6.
    Kapustyan, A.V., Melnik, V.S., Valero, J., Yasinsky, V.V.: Global Attractors of Multi-valued Dynamical Systems and Evolution Equations without Uniqueness. Naukova Dumka, Kyiv (2008)Google Scholar
  7. 7.
    Kapustyan, A.V., Valero, J.: Attractors of multivalued semiflows generated by differential inclusions and their approximations. Abstr. Appl. Anal. 5, 33–46 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kapustyan, A.V., Valero, J.: On the connectedness and asymptotic behaviour of solutions of reaction–diffusion systems. J. Math. Anal. Appl. 323, 614–633 (2006)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kapustyan, A.V., Valero, J.: Weak and strong attractors for the 3D Navier–Stokes system. J. Differ. Equ. 240, 249–278 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kloeden, P.E., Marín-Rubio, P.: Negatively invariant sets and entire trajectories of set-valued dynamical systems. Set-Valued Var. Anal. 19, 43–57 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Melnik, V.S., Valero, J.: On attractors of multivalued semi-flows and differential inclusions. Set-Valued Anal. 6, 83–111 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Melnik, V.S., Valero, J.: Addendum to “On attractors of multivalued semiflows and differential inclusions”. Set-Valued Anal. 16, 507–509 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Roxin, E.: Stability in general control systems. J. Differ. Equ. 1, 115–150 (1965)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Szegö, G.P., Treccani, G.: Semigruppi di Trasformazioni Multivoche. In: Springer Lecture Notes in Mathematics, vol. 101. Springer, Heidelberg (1969)Google Scholar
  15. 15.
    Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1979)Google Scholar
  16. 16.
    Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V.: Evolution Inclusions and Variational Inequalities for Earth Data Processing III. Springer, Berlin (2012)CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut für MathematikGoethe UniversitätFrankfurt am MainGermany
  2. 2.Departamento de Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevillaSpain
  3. 3.Universidad Miguel Hernandez de ElcheCentro de Investigación OperativaElcheSpain

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