Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations

Alain Miranville; Xiaoming Wang

doi:10.3934/dcds.1996.2.95

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1996, Volume 2, Issue 1: 95-110. Doi: 10.3934/dcds.1996.2.95

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Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations

Alain Miranville¹ and
Xiaoming Wang^2,

1.
Laboratoire d'Analyse Numérique, Université Paris-Sud, Bâtiment 425, 91405 Orsay
2.
Department of Mathematics, Indiana University, Bloomington, IN 47405

Received: April 30, 1995

Published: December 1995

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Abstract

Our aim in this article is to derive an upper bound on the dimension of the attractor for Navier-Stokes equations with nonhomogeneous boundary conditions. In space dimension two, for flows in general domains with prescribed tangential velocity at the boundary, we obtain a bound on the dimension of the attractor of the form $c\mathcal{R} e^{3/2}$, where $\mathcal{R} e$ is the Reynolds number. This improves significantly on previous bounds which were exponential in $\mathcal{R} e$.

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Mathematics Subject Classification: 35B30, 35B40, 35Q30, 76D05, 76F10, 76F20, 58F39, 58F13, 28A78, 28A80.

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