Exponential attractors for the slightly compressible 2D-Navier-Stokes

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July 1996, 2(3): 315-348. doi: 10.3934/dcds.1996.2.315

Exponential attractors for the slightly compressible 2D-Navier-Stokes

1.	Université Bordeaux-I, Mathématiques Appliquées, 351 Cours de la Libération, 33405 Talence Cedex
2.	Université de Bordeaux I, Laboratoire de Mathématiques Appliquées de Bordeaux, 351 Cours de la Libération, 33405 Talence Cedex

Received February 1996 Published May 1996

Abstract
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The equations of a slightly compressible fluid have been introduced to approach, when the parameter of compressibility $\epsilon$ is small, the incompressible Navier-Stokes equations. The object of this article is to prove the existence of exponential attractors in the 2D case for this partially dissipative system: The equations of a slightly compressible fluid. Furthermore, we establish an upper-bound of the fractal dimension of the exponential attractors described by the variable $(u^\epsilon, \sqrt{\epsilon}p^\epsilon)$; $u^\epsilon$ being the velocity and $p^\epsilon$ the pressure. Furthermore, a lower-semicontinuity result of these exponential attractors to the one of incompressible Navier-Stokes equations is obtained. These properties are linked to the existence of uniform absorbing sets with respect to $\epsilon$ for $\epsilon \leq \epsilon_0$ in the variable $(u^\epsilon, \sqrt{\epsilon}p^\epsilon)$ ($\epsilon_0$ fixed).

Keywords: Navier-Stokes equations, approximation., Exponential attractors.

Mathematics Subject Classification: 35B40, 35B45, 35B65, 35Q30, 35B3.

Citation: Pierre Fabrie, C. Galusinski. Exponential attractors for the slightly compressible 2D-Navier-Stokes. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 315-348. doi: 10.3934/dcds.1996.2.315

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