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 + ∂xu
3
+
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:





∂t
2u
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
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