Ill-posedness issues for nonlinear dispersive equations

TL;DR

This paper discusses the challenges of well-posedness in nonlinear dispersive equations and introduces recent methods for establishing ill-posedness through high-frequency approximate solutions.

Contribution

It presents new techniques for proving ill-posedness in dispersive PDEs, focusing on high-frequency solution constructions over small time intervals.

Findings

Ill-posedness can be demonstrated using high-frequency approximate solutions.

Recent methods effectively identify ill-posed regimes for dispersive PDEs.

Construction of solutions depends on frequency and small time intervals.

Abstract

These notes 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).