An effective model of the dynamics of a barotropic gas with fast oscillating initial data

Under study are the classical three-dimensional Navier–Stokes equations of a compressible inhomogeneous viscous fluid in a smooth bounded domain endowed with no-slip conditions on the boundary of the domain and fast oscillating initial density distributions. The state equation of the medium is the state equation for a barotropic gas. We assume that the adiabatic constant is greater than 3. We give a rigorous derivation of the homogenization procedure as the frequencies of fast oscillations tend to infinity and obtain a limit effective model of the dynamics of a compressible viscous gas with fast oscillating initial data.