  For a free surface problem of a highly subsonic heat-conducting inviscid
flow, motivated by a geometric approach developed by Christodoulou and Lindblad
in the study of the free surface problem of incompressible inviscid flows, the
a priori estimates of Sobolev norms in 2-D and 3-D are proved under the Taylor
sign condition by identifying a suitable higher order energy functional. The
estimates for some geometric quantities such as the second fundamental form and
the injectivity radius of the normal exponential map of the free surface are
also given. The novelty in our analysis includes dealing with the strong
coupling of large variation of temperature, heat-conduction, compressibility of
fluids and the evolution of free surface, loss of symmetries of equations, and
loss of derivatives in closing the argument which is a key feature compared
with Christodoulou and Lindblad's work. The motivation of this paper is to
contribute to the program of understanding the role played by the
heat-conductivity to free surface motions of inviscid compressible flows and
the behavior of such motions when the Mach number is small.
