Ill-posedness issues on $(abcd)$-Boussinesq system

TL;DR

This paper investigates the ill-posedness of the $(abcd)$-Boussinesq system, a model for small-amplitude surface waves, revealing parameter regimes where solutions are unstable, especially in two dimensions, based on high-to-low frequency cascades.

Contribution

It extends ill-posedness results to the general $(abcd)$-Boussinesq system, including optimal results for the two-dimensional BBM-BBM case, using frequency cascade analysis.

Findings

Identifies parameter regimes with ill-posedness.

Establishes optimal ill-posedness results in 2D.

Highlights the role of high-to-low frequency cascades.

Abstract

In this paper, we consider the Cauchy problem for $(abcd)$-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona, Chen, and Saut, describes a small-amplitude waves on the surface of an inviscid fluid, and derived as a first-order approximation of incompressible, irrotational Euler equations. We mainly establish the ill-posedness of the system under various parameter regimes, which generalize the result of the one-dimensional BBM-BBM case by Chen and Liu. Most of results established here, we obtain the optimal result for two-dimensional BBM-BBM system. The proof follows from an observation of the \emph{high to low-frequency cascade} present in nonlinearity, motivated by Bejenaru and Tao.