# Mathematics Colloquia and Seminars

For the 2d Euler equations, we provide an elementary constructive proof of shock formation for smooth solutions, having smooth initial datum of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data prescribed at time $t=t_0$, which has minimum slope $-1/\epsilon$, there exist smooth solutions to the Euler equations which form a shock at time $t=T_*$ with $T_*-t_0=O(\epsilon)$. The blowup time and location can be explicitly computed. The solution at the blowup time has Holder regularity $C^{1/3}$.