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

In this paper we study the Leray weak solutions of the incompressible Navier Stokes equation in an exterior do- main. We describe, in particular, a hyperbolic version of the so called artificial compressibility method investigated by J.L. Lions and Temam. The convergence of these type of approx- imations shows in general a lack of strong convergence due to the presence of acoustic waves. In this paper we face this diffi- culty by taking care of the dispersive nature of these waves by means of the Strichartz estimates or waves equations satisfied by the pressure. We introduce wave equations to take care of the pressure in different acoustic components, each one of them satisfying a specific initial boundary value problem. The strong convergence analysis of the velocity field will be achieved by us- ing the associated Leray-Hodge decomposition.