The APDE seminar on Monday, 2/12, will be given by Steve Shkoller (UC Davis) online via Zoom from 4:10pm to 5:00pm PST (in particular, there will be no in-person talk). To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: The geometry of maximal development and shock formation for the Euler equations

Abstract: We establish the maximal hyperbolic development of Cauchy data for the multi-dimensional compressible Euler equations throughout the shock formation process. For an open set of compressive and generic $H^7$ initial data, we construct unique $H^7$ solutions to the Euler equations in the maximal spacetime region such that at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-$2$ surface of “first singularities” called the pre-shock set; second, a downstream hypersurface emanating from the pre-shock set, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock set, which the Euler solution cannot reach. This talk is based on joint work with Vlad Vicol at NYU.