  For the 3D compressible isentropic Euler equations with an initial
perturbation of size $\ve$ of a rest state, if the initial vorticity is of size
$\dl$ with $0<\dl\le \ve$ and $\ve$ is small, we establish that the lifespan of
the smooth solutions is $T_{\dl}=O(\min\{e^\frac{1}{\ve},\frac{1}{\delta}\})$
for the polytropic gases, and $T_{\dl}=O(\frac{1}{\delta})$ for the Chaplygin
gases. For example, when $\dl=e^{-\f{1}{\ve^2}}$ is chosen, then
$T_{\dl}=O(e^{\f{1}{\ve}})$ for the polytropic gases and
$T_{\dl}=O(e^{\f{1}{\ve^2}})$ for the Chaplygin gases although the
perturbations of the initial density and the divergence of the initial velocity
are only of order $O(\ve)$. Our result illustrates that the time of existence
of smooth solutions depends crucially on the size of the vorticity of the
initial data, as long as the initial data is sufficiently close to a constant.
The main ingredients in the paper are: introducing some suitably weighted
energies, deriving the pointwise space-time decay estimates of solutions,
looking for the good unknown instead of the velocity, and establishing the
required weighted estimates on the vorticty.
