  This paper concerns the global well posedness issue of the Navier-Stokes
equations (CNS) describing barotropic compressible fluid flow with free surface
occupied in the three dimensional exterior domain. Combining the maximal
$L_p$-$L_q$ estimate and the $L_p$-$L_q$ decay estimate of solutions to the
linearized equations, we prove the unique existence of global in time solutions
in the time weighted maximal $L_p$-$L_q$ regularity class for some $p>2$ and
$q>3.$ Namely, the solution is bounded as $L_p$ in time and $L_q$ in space.
Compared with the previous results of the free boundary value problem of (CNS)
in unbounded domains, we relax the regularity assumption on the initial states,
which is the advantage by using the maximal $L_p$-$L_q$ regularity framework.
On the other hand, the equilibrium state of the moving boundary of the exterior
domain is not necessary the sphere. To our knowledge, this paper is the first
result on the long time solvability of the free boundary value problem of (CNS)
in the exterior domain.
