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In this paper we study the Leray weak solutions of the incompressible NavierStokes equation in an exterior domain.We describe, in particular, an hyperbolicversion of the so called artificial compressibility method investigated byJ.L.Lions and Temam. The convergence of these type of approximation show ingeneral a lack of strong convergence due to the presence of acoustic waves. Inthis paper we face this difficulty by taking care of the dispersive nature ofthese waves by means of the Strichartz estimates or waves equations satisfiedby the pressure. We actually decompose the pressure in different acousticcomponents, each one of them satisfies a specific initial boundary valueproblem. The strong convergence analysis of the velocity field will be achievedby using the associated Leray-Hodge decomposition.

Leray weak solutions of the Incompressible Navier Stokes system on exterior domains via the artificial compressibility method