  We concern with the global existence and large time behavior of compressible
fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an
infinitely expanding ball. Such a problem is one of the interesting models in
studying the theory of global smooth solutions to multidimensional compressible
gases with time dependent boundaries and vacuum states at infinite time. Due to
the conservation of mass, the fluid in the expanding ball becomes rarefied and
eventually tends to a vacuum state meanwhile there are no appearances of vacuum
domains in any part of the expansive ball, which is easily observed in finite
time. In this paper, as the first part of our three papers, we will confirm
this physical phenomenon for the compressible inviscid fluids by obtaining the
exact lower and upper bound on the density function.
