本文考虑一维和二维欧几里得空间上的$(abcd)$ -Boussinesq系统的Cauchy问题.该模型最初由Bona、Chen和Saut提出，描述了无粘流体表面上的小振幅波，并推导为不可压缩无旋欧拉方程的一阶近似。我们主要建立了系统在不同参数区域下的不适定性，推广了Chen和Liu关于一维BBM-BBM情形的结果。在此基础上，我们得到了二维BBM-BBM系统的最优解。证明如下从观察的\emph{high to low-frequency cascade}目前在非线性，动机Bejenaru和道。
英文摘要：
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.
Singularity formation for the Serre-Green-Naghdi equations and applications to abcd-Boussinesq systems
在这项工作中，我们证明了当界面切向到达不透水底部时，Serre-Green-Naghdi方程的解不能全局定义。因此，我们的结果是对\emph{Camassa, R., Falqui, G., Ortenzi, G., Pedroni, M., \& Thomson, C. Hydrodynamic models and confinement effects by horizontal boundaries. Journal of Nonlinear Science, 29(4), 1445-1498, 2019.}的补充.此外，我们还证明了$abcd-$ Boussinesq系统的解在有限时间内可以变号.最后，我们证明了$abcd-$ Boussinesq系统的有限时间奇异性.后者的数学结果与\emph{Bona, \& Chen, Singular solutions of a Boussinesq system for water waves. J. Math. Study, 49(3), 205-220, 2016}中的数值有关。
Strong ill-posedness of the 2D Boussinesq equations in supercritical Besov spaces
School of Mathematics and Computer Sciences, Gannan Normal University（赣南师范大学数学与计算机科学学院）；School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）
Rigorous numerics for ill-posed PDEs: periodic orbits in the Boussinesq equation
VU University Amsterdam（阿姆斯特丹自由大学）；Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo（圣保罗大学数学与计算科学学院）；Université Laval, Département de Mathématiques et de Statistique（拉瓦尔大学数学与统计系）
