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January  1996, 2(1): 95-110. doi: 10.3934/dcds.1996.2.95

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

Alain Miranville 1, and  Xiaoming Wang 2,

1. 

Laboratoire d'Analyse Numérique, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France
2. 

Department of Mathematics, Indiana University, Bloomington, IN 47405, United States

Received  May 1995 Published  October 1995

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 $\mathcal{R} e$.
Keywords: energy dissipation rate, Taylor-Couette flows, Hausdorff and fractal dimensions, global attractor, Reynolds number., channel flows, background flows, functional invariant set, Navier-Stokes equations.
Mathematics Subject Classification: 35B30, 35B40, 35Q30, 76D05, 76F10, 76F20, 58F39, 58F13, 28A78, 28A8.
Citation: Alain Miranville, Xiaoming Wang. Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 95-110. doi: 10.3934/dcds.1996.2.95
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