TY - JOUR
T1 - Small-data shock formation in solutions to 3D quasilinear wave equations
T2 - An overview
AU - Holzegel, Gustav
AU - Klainerman, Sergiu
AU - Speck, Jared
AU - Wong, Willie Wai Yeung
N1 - Funding Information:
We would like to thank the American Institute of Mathematics for funding three SQuaREs workshops on the formation of shocks, which greatly aided the development of many of the ideas presented in this paper. We would also like to thank Jonathan Luk and Shiwu Yang for participating in the workshops and for their helpful contributions, as well as Hans Lindblad for sharing his insight on Alinhac’s work. GH is grateful for the support offered by a grant of the European Research Council. SK is grateful for the support offered by NSF grant # DMS- 1362872. JS is grateful for the support offered by NSF grant # DMS-1162211 and by a Solomon Buchsbaum grant administered by the Massachusetts Institute of Technology. WW is grateful for the support offered by the Swiss National Science Foundation through a grant to Joachim Krieger.
Publisher Copyright:
© 2016 World Scientific Publishing Company.
PY - 2016/3/1
Y1 - 2016/3/1
N2 - 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.
AB - 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.
KW - Characteristic hypersurfaces
KW - Raychaudhuri equation
KW - Riccati equation
KW - compatible current
KW - eikonal function
KW - generalized energy estimates
KW - hyperbolic conservation laws
KW - maximal development
KW - null condition
KW - vectorfield method
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U2 - 10.1142/S0219891616500016
DO - 10.1142/S0219891616500016
M3 - Article
AN - SCOPUS:84961841609
SN - 0219-8916
VL - 13
SP - 1
EP - 105
JO - Journal of Hyperbolic Differential Equations
JF - Journal of Hyperbolic Differential Equations
IS - 1
ER -