An ill-posedness result for the Boussinesq equation

TL;DR

This paper establishes that the nonlinear 'good' Boussinesq equation is ill-posed in Sobolev spaces with regularity below -1/2, showing the flow map is not continuous in these spaces.

Contribution

It proves new ill-posedness results for the Boussinesq equation, demonstrating discontinuity of the flow map in Sobolev spaces for all s < -1/2.

Findings

Flow map is discontinuous in Sobolev spaces for s < -1/2

Results apply to both periodic and non-periodic problems

Advances understanding of regularity thresholds for the equation

Abstract

The aim of this article is to prove new ill-posedness results concerning the nonlinear "good" Boussinesq equation, for both the periodic and non-periodic initial value problems. Specifically, we prove that the associated flow map is not continuous in Sobolev spaces $H^s$, for all $s<-1/2$.