Small-data shock formation in solutions to 3D quasilinear wave equations: An overview
Name
Speck_Shock formation.pdf
Size
835.12 KB
Format
Adobe PDF
Checksum (MD5)
daf4685e28bd6c21795e13759021cae4
Author(s)
Holzegel, Gustav
Klainerman, Sergiu
Wong, Willie Wai-Yeung
Speck, Jared R.
Date Issued
March 2016
Journal
Journal of Hyperbolic Differential Equations
Publisher
World Scientific
Citation
Holzegel, Gustav et al. “Small-Data Shock Formation in Solutions to 3D Quasilinear Wave Equations: An Overview.” Journal of Hyperbolic Differential Equations 13.1 (2016): 1–105.
Version
Original manuscript
Abstract
In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small H[superscript s]-initial conditions (with ss 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1142/s0219891616500016
