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

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2010, 28(1): 161-179. doi: 10.3934/dcds.2010.28.161

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, Germany
2.	Centro de Investigation Operativa, Universidad Miguel Hernández, E-03202 Elche (Alicante), Spain

Received January 2009 Revised January 2010 Published April 2010

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

Mathematics Subject Classification: 35B40, 35B41, 35Q30, 37B25, 58C0.

Citation: Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161

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