We establish global existence of solutions to the compressible Euler
equations, in the case that a finite volume of ideal gas expands into vacuum.
Vacuum states can occur with either smooth or singular sound speed, the latter
corresponding to the so-called physical vacuum singularity when the enthalpy
vanishes on the vacuum wave front like the distance function. In this instance,
the Euler equations lose hyperbolicity and form a degenerate system of
conservation laws, for which a local existence theory has only recently been
developed. Sideris found a class of expanding finite degree-of-freedom
global-in-time affine solutions, obtained by solving nonlinear ODEs. In three
space dimensions, the stability of these affine solutions, and hence global
existence of solutions, was established by Had\v{z}i\'{c} \& Jang with the
pressure-density relation p=ργ with the constraint that 1<γ≤35. They asked if a different approach could go beyond
the γ>35 threshold. We provide an affirmative answer to
their question, and prove stability of affine flows and global existence for
all γ>1, thus also establishing global existence for the shallow water
equations when γ=2.Comment: 51 pages, details added to Section 4.7, to appear in Arch. Rational
Mech. Ana