  In a recent work Sideris constructed a finite-parameter family of compactly
supported affine solutions to the three-dimensional isentropic compressible
Euler equations satisfying the physical vacuum condition. The support of these
solutions expands at a linear rate in time. We show that if the adiabatic
exponent $\gamma$ belongs to the interval $(1,\frac53]$, then these affine
motions are nonlinearly stable. Small perturbations lead to globally-in-time
defined solutions that remain in the vicinity of the manifold of affine
motions, they remain smooth in the interior of their support, and no shocks are
formed in the process.
  Our strategy relies on two key ingredients. We first provide a new
interpretation of the affine motions using an (almost) invariant action of
GL$^+(3)$ on the compressible Euler system. This transformation dictates a
particular rescaling of time and a change of variables, which in turn exposes a
stabilization mechanism induced by the expansion of the background affine
motions when $\gamma\in(1,\frac53]$. We then switch to a Lagrangian description
of the rescaled Euler system which reflects the geometry of expanding affine
motions. We introduce new ideas with respect to the existing well-posedness
frameworks and build high-order energy spaces to prove global-in-time
stability, thereby making crucial use of the new stabilization effect
