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

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  4. Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. American Mathematical Society, Providence, RI (2002)

    MATH  Google Scholar 

  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. 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)

  7. Kapustyan, A.V., Valero, J.: Attractors of multivalued semiflows generated by differential inclusions and their approximations. Abstr. Appl. Anal. 5, 33–46 (2000)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kapustyan, A.V., Valero, J.: Weak and strong attractors for the 3D Navier–Stokes system. J. Differ. Equ. 240, 249–278 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Melnik, V.S., Valero, J.: On attractors of multivalued semi-flows and differential inclusions. Set-Valued Anal. 6, 83–111 (1998)

    Article  MathSciNet  Google Scholar 

  12. Melnik, V.S., Valero, J.: Addendum to “On attractors of multivalued semiflows and differential inclusions”. Set-Valued Anal. 16, 507–509 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Roxin, E.: Stability in general control systems. J. Differ. Equ. 1, 115–150 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Szegö, G.P., Treccani, G.: Semigruppi di Trasformazioni Multivoche. In: Springer Lecture Notes in Mathematics, vol. 101. Springer, Heidelberg (1969)

    Google Scholar 

  15. Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1979)

  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)

    Book  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Goethe Universität, 60054, Frankfurt am Main, Germany

    P. E. Kloeden

  2. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080, Sevilla, Spain

    P. Marín-Rubio

  3. Universidad Miguel Hernandez de Elche, Centro de Investigación Operativa, Avda. Universidad s/n, 03202, Elche, Spain

    J. Valero

Authors
  1. P. E. Kloeden
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  2. P. Marín-Rubio
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  3. J. Valero
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Correspondence to J. Valero.

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Kloeden, P.E., Marín-Rubio, P. & Valero, J. The Envelope Attractor of Non-strict Multivalued Dynamical Systems with Application to the 3D Navier–Stokes and Reaction–Diffusion Equations. Set-Valued Var. Anal 21, 517–540 (2013). https://doi.org/10.1007/s11228-012-0228-x

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  • DOI: https://doi.org/10.1007/s11228-012-0228-x

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