  In the presence of vacuum, the physical entropy for polytropic gases behaves
singularly and it is thus a challenge to study its dynamics. It is shown in
this paper that the boundedness of the entropy can be propagated up to any
finite time provided that the initial vacuum presents only at far fields with
sufficiently slow decay of the initial density. More precisely, for the Cauchy
problem of the one dimensional heat conductive compressible Navier-Stokes
equations, the global well-posedness of strong solutions and uniform
boundedness of the corresponding entropy are established, as long as the
initial density vanishes only at far fields with a rate no more than
$O(\frac{1}{x^2})$. The main tools of proving the uniform boundedness of the
entropy are some singularly weighted energy estimates carefully designed for
the heat conductive compressible Navier-Stokes equations and an elaborate De
Giorgi type iteration technique for some classes of degenerate parabolic
equations. The De Giorgi type iterations are carried out to different equations
in establishing the lower and upper bounds of the entropy.
