# American Institute of Mathematical Sciences

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

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).
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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