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).
Bibliographical noteFunding Information:
C. Kwak was partially supported by Project France-Chile ECOS-Sud C18E06, and is partially supported by the Ewha Womans University Research Grant of 2020 and the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (No. 2020R1F1A1A0106876811).
Ch.M. was partially funded by Chilean research Grants FONDECYT 1191412, CONICYT PFCHA/DOCTORADO NACIONAL/2016-21160593 and CMM ANID PIA AFB170001.
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- Boussinesq system