  In this paper, we study the low Mach and Froude number limit for the all-time
classical solution of a fluid-vacuum free boundary problem of one-dimensional
compressible Navier-Stokes equations. No smallness of initial data for the
existence of all-time solutions are supposed. The uniform estimates of
solutions with respect to the Mach number and the Froude number are established
for all the time, in particular for high order derivatives of the pressure,
which is a novelty in contrast to previous results. The cases of "ill-prepared"
initial data and "well-prepared" initial data are both discussed. It is
interesting to see, either both the Mach number and the Froude number vanish,
or the time goes to infinity, the limiting functions are the same, that is, the
steady state. The main difficulty is that, the system is degenerate near the
free boundary and contains singular terms. This result can be viewed as the
first one on the low Mach and Froude numbers limit for free boundary problems.
At the same time, we also establish the all-time existence of the classical
solution with sharp convergent rates to the steady state, while previous
results are only concerned with the weak or strong solutions.
