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

K¨ 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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