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SUMMARY:Advances in the theory of multi-dimensional shock waves - Jared Sp
 eck (Vanderbilt University)
DTSTART:20221114T140000Z
DTEND:20221114T150000Z
UID:TALK192236@talks.cam.ac.uk
CONTACT:Zexing Li
DESCRIPTION:A shock singularity in a quasilinear hyperbolic PDE solution i
 s a mild singularity such that one of the solution’s derivatives blows u
 p\, though the solution itself remains bounded. Importantly\, the mild nat
 ure of the singularity opens the door to the possibility that the solution
  might be continued uniquely as a weak solution past the singularity\, und
 er suitable selection criteria. While the rigorous 1D theory is in a matur
 e stage due to the availability of well-posedness results for BV initial d
 ata\, multi-dimensional hyperbolic PDEs are typically ill-posed in BV. Con
 sequently\, the theory of multi-dimensional shocks is permeated with funda
 mental open problems\, many with deep ties to geometry. Despite the challe
 nges in higher dimensions\, for specific systems\, including the compressi
 ble Euler equations and relativistic Euler equations in 3D\, there has bee
 n dramatic progress in the last 15 years\, starting with Christodoulou’s
  2007 monograph on shock formation in irrotational solutions. In this talk
 \, after providing an introduction to the 1D problem\, I will give a non-t
 echnical description of recent advances in multi-dimensions\, with a focus
  on the multi-dimensional compressible Euler equations with vorticity and 
 entropy. Many recent results are based on a new formulation of compressibl
 e Euler flow exhibiting miraculous geo-analytic structures and regularity 
 properties\, and the analysis fundamentally relies on nonlinear geometric 
 optics. In particular\, I will describe my recent series of works on the 3
 D compressible Euler equations with vorticity and entropy\, which\, for op
 en sets of initial data\, reveal the full structure of the maximal classic
 al development\, including the full structure of the singular set as well 
 the emergence of a Cauchy horizon from the singularity. Finally\, time per
 mitting\, I will discuss some of the many open problems in the field. Vari
 ous aspects of this program are joint with L. Abbrescia\, J. Luk\, and M. 
 Disconzi.
LOCATION:CMS\, MR13
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