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.