  We consider one-dimensional self-similar solutions to the isentropic Euler
system when the initial data are at vacuum to the left of the origin. For $x>0$
the initial velocity and sound speed are of form $u_0(x)=u_+x^{1-\lambda}$ and
$c_0(x)=c_+x^{1-\lambda}$, for constants $u_+\in\RR$, $c_+>0$, $\lambda\in\RR$.
We analyze the resulting solutions in terms of the similarity parameter
$\lambda$, the adiabatic exponent $\gamma$, and the initial (signed) Mach
number $\text{Ma}=u_+/c_+$. Restricting attention to locally bounded data, we
find that when the sound speed initially decays to zero in a H\"older manner
($0<\lambda<1$), the resulting flow is always defined globally. Furthermore,
there are three regimes depending on $\text{Ma}$: for sufficiently large
positive $\text{Ma}$-values, the solution is continuous and the initial
H\"older decay is immediately replaced by $C^1$-decay to vacuum along a
stationary vacuum interface; for moderate values of $\text{Ma}$, the solution
is again continuous and with an accelerating vacuum interface along which $c^2$
decays linearly to zero (i.e., a "physical singularity''); for sufficiently
large negative $\text{Ma}$-values, the solution contains a shock wave emanating
from the initial vacuum interface and propagating into the fluid, together with
a physical singularity along an accelerating vacuum interface. In contrast,
when the sound speed initially decays to zero in a $C^1$ manner ($\lambda<0$),
a global flow exists only for sufficiently large positive values of
$\text{Ma}$. Non-existence of global solutions for smaller $\text{Ma}$-values
is due to rapid growth of the data at infinity and is unrelated to the presence
of a vacuum.
