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 et al. (J Nonlinear Sci 12:283-318, 2002, Nonlinearity 17:925-952, 2004), describes a small-amplitude waves on the surface of an inviscid fluid, and is 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 one-dimensional BBM-BBM case by Chen and Liu (Anal Math 121:299-316, 2013). Among results established here, we emphasize that the ill-posedness result for two-dimensional BBM-BBM system is optimal. The proof follows from an observation of the high to low frequency cascade present in nonlinearity, motivated by Bejenaru and Tao (J Funct Anal 233:228-259, 2006).