VAPS 56:"The structure of the maximal development in 3D compressible Euler flow"

Speaker: Leonardo Abbrescia, Vanderbilt and Georgia Tech
Abstract: will discuss my works with J. Speck on 3D compressible Euler flow, in which we reveal the structure of the maximal Cauchy (MCD) development for open sets of initial data without symmetry, irrotationality, or isentropicity assumptions. The MCD is roughly the largest spacetime region on which the solution exists classically and is uniquely determined by the initial data - the holy grail in the classical study of hyperbolic equations. In particular, we describe the full structure of the singular set, where the solution’s gradient blows up in a shock singularity, as well as the emergence of a Cauchy horizon from the singularity. It is known that MCD's might fail to be unique unless one constructs it in its entirety and proves that it enjoys some crucial properties. The portion of the MCDs we construct proves that a localized version of the crucial properties hold for shock-forming initial data. Time permitting, I will discuss some of the many open problems in the field.