Introduction In the past 25 years or so, there has been considerable interest in the study of nonlinear partial differential equations, modeling phenomena of wave propagation, coming from physics and engineering. The areas that gave rise to these equations are water waves, optics, lasers, ferromagnetism, general relativity, sigma models and many others. These equations also have connections to geometric flows and to ahler and Minkowski geometries. Examples of such equations are the generalized KdV equations: ∂tu + ∂3u x + uk∂xu = 0,x R,t R u|t=0 = u0, the nonlinear Schr¨ odinger equations: i∂tu + Δu ± |u|pu = 0,x RN,t R u|t=0 = u0, and the nonlinear wave equation: ∂2u t Δu = ±|u|pu, x RN,t R u|t=0 = u0 ∂tu|t=0 = u1. Inspired by the theory of ODE one defines a notion of well-posedness for these initial value problems (IVP), with data u0 ((u0,u1)) in a given function space B. Since these equations are time reversible, the intervals of time to be considered are symmetric around the origin. Well-posedness entails existence, uniqueness of a solution which describes a continuous curve in the space B, for t I, the interval of existence, and continuous dependence of the curve on the initial data. If I is finite we call this local well-posedness (lwp) if I is the whole line, we call this global well-posedness (gwp). The first stage of development of the theory concentrated on the “local theory of the Cauchy problem”, which established local well-posedness results on Sobolev spaces B, or global well-posedness for small data in B. Pioneering works were due to Segal, Strichartz, Kato, Ginibre-Velo, Pecher and many others. In the late 80’s, in collaboration with Ponce and Vega we introduced the systematic use of the machinery of modern harmonic analysis to study the “local theory of the Cauchy problem”. Further contributions came from work of Bourgain, Klainerman- Machedon, Tataru, Tao and many others. In recent years, there has been a lot of interest in the study, for nonlinear dispersive equations, of the long-time behaviour of solutions, for large data. Issues like blow-up, global existence and scattering have come to the forefront, especially in critical problems. These problems are natural extensions of nonlinear elliptic problems, which were studied earlier. To explain this connection, recall that in vii
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