The Kneser property of the weak solutions of the three  dimensional Navier-Stokes equations

Peter E. Kloeden; José Valero

doi:10.3934/dcds.2010.28.161

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2010, Volume 28, Issue 1: 161-179. Doi: 10.3934/dcds.2010.28.161

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The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations

Peter E. Kloeden^1, and
José Valero^2,

1.
Institut für Mathematik, Johann Wolfgang Goethe Universität, D-60054 Frankfurt am Main
2.
Centro de Investigation Operativa, Universidad Miguel Hernández, E-03202 Elche (Alicante)

Received: December 31, 2008

Revised: December 31, 2009

Published: February 2010

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Abstract

The Kneser theorem for ordinary differential equations without uniqueness says that the attainability set is compact and connected at each instant of time. We establish corresponding results for the attainability set of weak solutions for the 3D Navier-Stokes equations satisfying an energy inequality. First, we present a simplified proof of our earlier result with respect to the weak topology in the space $H$. Then we prove that this result also holds with respect to the strong topology on $H$ provided that the weak solutions satisfying the weak version of the energy inequality are continuous. Finally, using these results, we show the connectedness of the global attractor of a family of setvalued semiflows generated by the weak solutions of the NSE satisfying suitable properties.

Keywords:

Attainability set,
Navier-Stokes equations,
globally modified Navier-Stokes Equations,
Kneser property,
weak connectedness,
weak compactness,
global weak attractor.

Mathematics Subject Classification: 35B40, 35B41, 35Q30, 37B25, 58C06.

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