Solubility of unsteady equations of multi-component viscous compressible fluids

We consider an initial-boundary value problem describing unsteady barotropic motions of multi-component mixtures of viscous compressible fluids in a bounded three-dimensional domain. The material derivative operator is assumed to be common for all components and determined by the average velocity of the mixture, but all other terms contain the separate velocities of the components. The pressure is assumed to be common and dependent on the total density. We impose no simplifying assumptions (in particular, on the structure of the viscosity matrix) besides those stated above and thus preserve all the terms in the equations that naturally extend the Navier–Stokes model of motions of one-component media. We prove the existence of weak generalized solutions of the initial-boundary value problem.