Abstract

A shock singularity in a quasilinear hyperbolic PDE solution is a mild singularity such that one of the solution’s derivatives blows up, though the solution itself remains bounded. Importantly, the mild nature of the singularity opens the door to the possibility that the solution might be continued uniquely as a weak solution past the singularity, under suitable selection criteria. While the rigorous 1D theory is in a mature stage due to the availability of well-posedness results for BV initial data, multi-dimensional hyperbolic PDEs are typically ill-posed in BV. Consequently, the theory of multi-dimensional shocks is permeated with fundamental open problems, many with deep ties to geometry. Despite the challenges in higher dimensions, for specific systems, including the compressible Euler equations and relativistic Euler equations in 3D, there has been 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-technical 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 compressible 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 3D compressible Euler equations with vorticity and entropy, which, for open sets of initial data, reveal the full structure of the maximal classical development, including the full structure of the singular set as well the emergence of a Cauchy horizon from the singularity. Finally, time permitting, I will discuss some of the many open problems in the field. Various aspects of this program are joint with L. Abbrescia, J. Luk, and M. Disconzi.