Ill-posedness results for generalized Boussinesq equations

TL;DR

This paper establishes ill-posedness results for generalized Boussinesq equations, demonstrating that the flow map lacks smoothness in certain Sobolev spaces, which indicates the limitations of standard solution methods.

Contribution

The authors extend previous ill-posedness results to generalized Boussinesq equations, identifying regularity thresholds for well-posedness.

Findings

Flow map is not smooth for certain Sobolev indices

Provides thresholds for regularity in solving Boussinesq equations

Extends previous results to generalized versions

Abstract

In this article we present ill-posedness results for generalized Boussinesq equations, which incorporate also the ones obtained by the authors for the classical "good" Boussinesq equation (arXiv:1202.6671). More precisely, we show that the associated flow map is not smooth for a range of Sobolev indices, thus providing a threshold for the regularity needed to perform a Picard iteration for these problems.