Formation of shocks for 2D isentropic compressible Euler

Understand

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.