Existence of weak solutions to the three-dimensional problem of steady barotropic motions of mixtures of viscous compressible fluids

We consider the boundary value problem describing the steady barotropic motion of a multicomponent mixture of viscous compressible fluids in a bounded three-dimensional domain. We assume that the material derivative operator is common to all components and is defined by the average velocity of the motion, but keep separate velocities of the components in other terms. Pressure is common and depends on the total density. Beyond that we make no simplifying assumptions, including those on the structure of the viscosity matrix; i.e., we keep all terms in the equations, which naturally generalize the Navier–Stokes model of the motion of one-component media. We establish the existence of weak solutions to the boundary value problem.