Formation of Point Shocks for 3D Compressible Euler

We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev-class initial data that are a small L^∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point, where they are of cusp-type with Hölder C^1/3 regularity. Our proof is based on the use of modulated self-similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self-similar variables.