Shock Formation in Small-Data Solutions to $3D$ Quasilinear Wave Equations: An Overview

TL;DR

This paper reviews Christodoulou's 2007 groundbreaking work on shock formation in small-data solutions to 3D quasilinear wave equations, highlighting new ideas and recent extensions that connect shock formation to the null condition.

Contribution

It provides an overview of Christodoulou's results, explains their main innovations, and discusses recent work extending these results to broader classes of equations.

Findings

Shock formation occurs for small-data solutions in 3D quasilinear wave equations.

Recent work shows shock formation relates to failure of the null condition.

The overview connects historical developments with recent advances.

Abstract

In his 2007 monograph, D. Christodoulou proved a remarkable result giving a detailed description of shock formation, for small $H^s$-initial conditions ($s$ sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by F. John in the mid 1970's and continued by S. Klainerman, T. Sideris, L. H\"ormander, H. Lindblad, S. Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of J. Speck, which extends Christodoulou's result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail the classic null condition.