Stationary solutions of a boundary value problem for equations of barotropic flow of multicomponent media

The asymptotic behavior (as $t\rightarrow +\infty$) of the solution to the initial-boundary value problem is analyzed for the system of differential equations describing the barotropic dynamics of a viscous  multifluid with a non-diagonal, symmetric and positive definite viscosity matrix, in the case of one spatial variable. New a priori estimates are obtained and stabilization of the solution to the initial-boundary value problem is proved.