Stabilization rate and stability for viscous compressible barotropic symmetric flows with free boundary for a general mass force

We consider symmetric flows of a viscous compressible barotropic fluid with free boundary driven by a general mass force $f_S$ (depending on both the Eulerian and the Lagrangian coordinates) and an outer pressure $p_{\Gamma,S}$, for a general monotone state function $p$. The case of self-gravitation arising in astrophysics is covered. Studied first are the existence, the uniqueness, and the static stability of positive stationary solutions; a variational study of these solutions and their static stability in terms of potential energy is presented. In the astrophysical context it is proved that the stationary solution is unique and statically stable, provided that the first adiabatic exponent is at least 4/3.
Next, in the case when the $\omega$-limit set for the non-stationary density and free boundary contains a statically stable positive stationary solution a uniform stabilization to this solution is deduced and, as the main result, stabilization-rate bounds of exponential type as $t\to\infty$ in $L^2$ and $H^1$ for the density and the velocity are established by constructing new non-trivial Lyapunov functionals for the problem. Moreover, it is proved that statically stable stationary solutions are exponentially asymptotically stable, and this non-linear dynamic stability is in addition stable with respect to small non-stationary perturbations of $f_S$ and $p_{\Gamma,S}$. A variational condition for the stationary solution is also introduced, which ensures global (with respect to the data) dynamic stability. The study is accomplished in the Eulerian coordinates and in the Lagrangian mass coordinates alike.