The study of water waves has a long history starting from Euler in 1752, and continues to be a very active area to the present day. Mathematically, the water wave equations describe the motion of water bounded above by a free surface. This free surface is subject to a constant (atmospheric) pressure, while gravity acts as an external force.
In this talk, I will start by demonstrating the underlying complexity of the physical system, and then I will discuss possible simplifications in the "shallow water" regime along with the relevant physical phenomena. In particular, I will focus on the singularity formation of the Cauchy problem for the simplified nonlocal shallow-water models, such as Camassa-Holm-type equations in 1D and 2D cases.