Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

Gustav Holzegel, Sergiu Klainerman, Jared Speck, Willie Wai Yeung Wong

Research output: Contribution to journalArticle

13 Scopus citations

Abstract

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small Hs-initial conditions (with 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 John in the mid 1970's and continued by Klainerman, Sideris, Hörmander, Lindblad, 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 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 to satisfy the classic null condition.

Original languageEnglish (US)
Pages (from-to)1-105
Number of pages105
JournalJournal of Hyperbolic Differential Equations
Volume13
Issue number1
DOIs
StatePublished - Mar 1 2016

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics(all)

Keywords

  • Characteristic hypersurfaces
  • Raychaudhuri equation
  • Riccati equation
  • compatible current
  • eikonal function
  • generalized energy estimates
  • hyperbolic conservation laws
  • maximal development
  • null condition
  • vectorfield method

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