  We consider the 2D isentropic compressible Euler equations, with pressure law
$p(\rho) = (\sfrac{1}{\gamma}) \rho^\gamma$, with $\gamma >1$. We provide an
elementary constructive proof of shock formation from smooth initial datum of
finite energy, with no vacuum regions, and with {nontrivial vorticity}. We
prove that for initial data which has minimum slope $- {\sfrac{1}{ \eps}}$, for
$ \eps>0$ taken sufficiently small relative to the $\OO(1)$ amplitude, there
exist smooth solutions to the Euler equations which form a shock in time
$\OO(\eps)$. The blowup time and location can be explicitly computed and
solutions at the blowup time are of cusp-type, with H\"{o}lder $C^
{\sfrac{1}{3}}$ regularity.
  Our objective is the construction of solutions with inherent $\OO(1)$
vorticity at the shock. As such, rather than perturbing from an irrotational
regime, we instead construct solutions with dynamics dominated by purely
azimuthal wave motion. We consider homogenous solutions to the Euler equations
and use Riemann-type variables to obtain a system of forced transport
equations. Using a transformation to modulated self-similar variables and
pointwise estimates for the ensuing system of transport equations, we show the
global stability, in self-similar time, of a smooth blowup profile.
