scholarly journals A note on quasilinear wave equations in two space dimensions

2015 ◽  
Vol 36 (5) ◽  
pp. 2855-2871 ◽  
Author(s):  
Dongbing Zha
Keyword(s):  
Wave Equations ◽  
Two Space Dimensions ◽  
Quasilinear Wave Equations

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

2016 ◽  
Vol 13 (01) ◽  
pp. 1-105 ◽  
Author(s):  
Gustav Holzegel ◽  
Sergiu Klainerman ◽  
Jared Speck ◽  
Willie Wai-Yeung Wong
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Large Body ◽  
Small Data ◽  
Shock Formation ◽  
Time Behavior ◽  
Long Time ◽  
Remarkable Result ◽  
Quasilinear Wave Equations ◽  
Three Space

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small [Formula: see text]-initial conditions (with [Formula: see text] 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.

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