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

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.