  Global solutions to the compressible Euler equations with heat transport by
convection in the whole space are shown to exist through perturbations of
Dyson's isothermal affine solutions. This setting presents new difficulties
because of the vacuum at infinity behavior of the density. In particular, the
perturbation of isothermal motion introduces a Gaussian function into our
stability analysis and a novel finite propagation result is proven to handle
potentially unbounded terms arising from the presence of the Gaussian. Crucial
stabilization-in-time effects of the background motion are mitigated through
the use of this finite propagation result however and a careful use of the heat
transport formulation in conjunction with new time weight manipulations are
used to establish global existence. The heat transport by convection offers
unique physical insights into the model and mathematically, we use a controlled
spatial perturbation in the analysis of this feature of our system which leads
us to exploit source term estimates as part of our techniques.
