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This paper is a continuation of the works in \cite{Euler} and \cite{NS},where the authors have established the global existence of smooth compressibleflows in infinitely expanding balls for inviscid gases and viscid gases,respectively. In this paper, we are concerned with the global existence andlarge time behavior of compressible Boltzmann gases in an infinitely expandingball. Such a problem is one of the interesting models in studying the theory ofglobal smooth solutions to multidimensional compressible gases with timedependent boundaries and vacuum states at infinite time. Due to theconservation of mass, the fluid in the expanding ball becomes rarefied andeventually tends to a vacuum state meanwhile there are no appearances of vacuumdomains in any part of the expansive ball, which is easily observed in finitetime. In the present paper, we will confirm this physical phenomenon for theBoltzmann equation by obtaining the exact lower and upper bound on themacroscopic density function.

The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, III: the 3-D Boltzmann equation