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

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).