Leonardo Abbrescia (Vanderbilt)

The APDE seminar on Monday, 11/14, will be given by Leonardo Abbrescia (Vanderbilt) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: A localized picture of the maximal development for shock forming solutions of the 3D compressible Euler equations

Abstract: It is well known that solutions to the inviscid Burgers’ equation form shock singularities in finite time, even when launched from smooth data. A far less documented fact, at least in the popular works on 1D hyperbolic conservation laws, is that shock singularities are intimately tied to a lack-of-uniqueness for the classical Burgers’ equation.

We prove that, locally, solutions to the Compressible Euler equations do not suffer from the same lack-of-uniqueness, even though they can be written as a coupled system of Burgers’ in isentropic plane-symmetry. Roughly, the saving grace is that Euler flow involves two speeds of propagation, and one of them “prevents” the mechanism driving the lack-of-uniqueness. Analytically, this is done by explicitly constructing a portion of the boundary of classical hyperbolic development for shock forming data. This boundary is a connected co-dimension 1 submanifold of Cartesian space, and we will discuss the delicate geo-analytic degeneracies and difficulties involved in its construction. This is joint work with Jared Speck.