On a stokes-like system for mixtures of fluids

We consider a simplified model that describes steady flows of a miscible mixture of fluids. The corresponding system of equations is studied in the whole space $\R^3$. The densities $\rho_i$ of the species and their velocity fields $\ui$ are prescribed at infinity: $\rho_i|_{\infty} = \rho_{i \infty} > 0\,,\;\; \ui|_{\infty} = 0$. We prove the existence of weak solutions to the system under consideration.