Global Behavior of 1d-Viscous Compressible Barotropic Fluid with a Free Boundary and Large Data

In the Eulerian coordinates we study 1D-flow of a viscous compressible barotropic fluid with an unknown free boundary for large initial data and mass force. Under fairly general conditions on the pressure function, viscosity coefficient, and a relation between the mass force and outer pressure we give the uniform with respect to time bounds for the solution and study its convergence to a stationary one as time tends to infinity. Moreover, in the case of uniquely defined stationary solution with strictly positive density we prove L-2- and H-1-stabilization rate estimates by constructing new Lyapunov functionals.