## Nikolay Tzvetkov: Nonlinear Dispersive Equations, Expository Lectures

### Ill - posedness issues for nonlinear dispersive equations

### Abstract

These lectures are devoted to the notion of well-posedness of the Cauchy
problem for nonlinear dispersive equations.
We present recent methods for proving ill-posedness type results for
dispersive PDE's.
The common feature in the analysis is that the proof of such results
requires the construction of high frequency
approximate solutions on small time intervals (possibly depending on the
frequency).

The classical notion of well-posedness, going back to Hadamard, requires
the existence, the uniqueness and the continuity
of the flow map on the spaces where the existence is established. It turns
out that in many cases a stronger form
of well-posedness holds. Namely, the flow map enjoys better continuity
properties as for example being Lipschitz continuous
on bounded sets. In such a situation we say that the corresponding problem
is semi-linearly well-posed in the corresponding
functional setting. Our main message is that for dispersive PDE's,
contrary to the case of hyperbolic PDE's,
the verification whether an equation in hand is semi-linearly well-posed
in a given functional framework requires a considerable care.

Our examples are KdV type equations and non linear Schrödinger
equations.