Shock formation for the Euler equations
Shock formation for the Euler equations
I will discuss a recent result with Tristan Buckmaster (Princeton) and Vlad Vicol (NYU) which establishes the finite-time听shock formation for the 3d isentropic Euler equations with vorticity. 听听We prove that for an open set of Sobolev-class initial data, there exist smooth solutions to the Euler equations which form a听generic听stable shock in finite time. The blow up time and location can be explicitly computed, and solutions at the blow up time are smooth except for听a single point, where they are of cusp-type with Holder听C1{3听regularity. Our proof is based on the use of modulated self-similar variables that are used to enforce a number of constraints on the blow up profile, necessary to establish the stability in self-similar variables of the generic shock profile.