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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 Schr\"odinger (NLS), wave maps (WM), Schr\"odinger maps (SM), generalised Korteweg-de Vries (gKdV), Maxwell-Klein-Gordon (MKG), and Yang-Mills (YM) equations. The classification of the nonlinearity as \emph{subcritical} (weaker than the linear dispersion at high frequencies), \emph{critical} (comparable to the linear dispersion at all frequencies), or \emph{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.

Global behaviour of nonlinear dispersive and wave equations