Set-Valued and Variational Analysis

, Volume 21, Issue 3, pp 517–540

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

Article

DOI: 10.1007/s11228-012-0228-x

Cite this article as:
Kloeden, P.E., Marín-Rubio, P. & Valero, J. Set-Valued Var. Anal (2013) 21: 517. doi:10.1007/s11228-012-0228-x

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.

Keywords

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 

Copyright information

© 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