A wave-breaking result for azimuthally varying water flows in cylindrical coordinates

We address here a question of fundamental importance in the analysis of nonlinear partial differential equations: when does a solution to a nonlinear partial differential equation develop singularities and what is the nature of those singularities? The particular type of singularity that we attend to here is wave breaking which is defined as the situation when the wave remains bounded up to the maximal existence time at which its slope becomes infinite. More specifically, our wave breaking result concerns the geophysical nonlinear water wave problem for an inviscid, incompressible, homogeneous fluid, written in cylindrical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions.