Analysis/Mathematical Physics Seminar

We will discuss a recent result with Tristan Buckmaster (Princeton) and Steve Shkoller (UC Davis) which establishes the finite-time shock formation for the 3d isentropic Euler equations with vorticity. We prove that for an open set of Sobolev-class initial data, there exist smooth solutions to the Euler equations which form a generic 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 a single point, where they are of cusp-type with Holder $C^{1/3}$ regularity. Our proof is based on the use of modulated self-similar variables that are used to enforce a number of constraints on the blow up profile, necessary to establish the stability in self-similar variables of the generic shock profile.