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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document}- and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{1}$\end{document}-stabilization rate estimates by constructing new Lyapunov functionals.