Ad hoc test functions for homogenization of compressible viscous fluid with application to the obstacle problem in dimension two

In this paper we highlight a set of ad hoc test functions to study the homogenization of viscous compressible fluid in domains with tiny holes. This set of functions allows to improve previous results in dimensions two and three. As an application we show that the presence of a small obstacle does not influence the dynamics of a viscous compressible fluid in dimension two.