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VOL. 2006 | 2008 Global behaviour of nonlinear dispersive and wave equations
Terence Tao

Editor(s) David Jerison, Barry Mazur, Tomasz Mrowka, Wilfried Schmid, Richard P. Stanley, Shing-Tung Yau

Abstract

We survey recent advances in the analysis of the large data global (and asymptotic) behaviour of nonlinear dispersive equations such as the nonlinear wave (NLW), nonlinear Schrödinger (NLS), wave maps (WM), Schrödinger maps (SM), generalised Korteweg-de Vries (gKdV), Maxwell-Klein-Gordon (MKG), and Yang-Mills (YM) equations. The classification of the nonlinearity as subcritical (weaker than the linear dispersion at high frequencies), critical (comparable to the linear dispersion at all frequencies), or supercritical (stronger than the linear dispersion at high frequencies) is fundamental to this analysis, and much of the recent progress has pivoted on the case when there is a critical conservation law. We discuss how one synthesises a satisfactory critical (scale-invariant) global theory, starting the basic building blocks of perturbative analysis, conservation laws, and monotonicity formulae, but also incorporating more advanced (and recent) tools such as gauge transforms, concentration-compactness, and induction on energy.

Information

Published: 1 February 2008
First available in Project Euclid: 10 October 2008

zbMATH: 1171.35004
MathSciNet: MR2459308

Rights: Copyright © 2008 International Press of Boston

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