The Tartar equation for homogenization of a model of the dynamics of fine-dispersion mixtures

We consider a mathematical model describing a nonstationary Stokes flow in a fine-dispersion mixture of viscous incompressible fluids with rapidly oscillating initial data. We perform homogenization of the model as the frequency of oscillations tends to infinity; this leads to the problem of finding effective coefficients of the averaged equations. To solve this problem, we propose and implement a method which bases on supplementing the averaged system with the Cauchy problem for the kinetic Tartar equation whose unique solution is the Tartar $H$-measure. Thereby we construct a correct closed model for describing the motion of a homogeneous mixture, because the effective coefficients of the averaged equations are uniquely expressed in terms of the $H$-measure.