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We consider the 2D isentropic compressible Euler equations, with pressure law$p(\rho) = (\sfrac{1}{\gamma}) \rho^\gamma$, with $\gamma >1$. We provide anelementary constructive proof of shock formation from smooth initial datum offinite energy, with no vacuum regions, and with {nontrivial vorticity}. Weprove that for initial data which has minimum slope $- {\sfrac{1}{ \eps}}$, for$ \eps>0$ taken sufficiently small relative to the $\OO(1)$ amplitude, thereexist smooth solutions to the Euler equations which form a shock in time$\OO(\eps)$. The blowup time and location can be explicitly computed andsolutions 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 irrotationalregime, we instead construct solutions with dynamics dominated by purelyazimuthal wave motion. We consider homogenous solutions to the Euler equationsand use Riemann-type variables to obtain a system of forced transportequations. Using a transformation to modulated self-similar variables andpointwise estimates for the ensuing system of transport equations, we show theglobal stability, in self-similar time, of a smooth blowup profile.

Formation of shocks for 2D isentropic compressible Euler