Large scale dynamics of the oceans and the atmosphere are commonly governed by the primitive equations (PEs), also known as Hydrostatic Euler Equations. It is well-known that the 3D viscous primitive equations are globally well-posed in Sobolev spaces. In this talk, first, I
will briefly discuss the ill-posedness in Sobolev spaces, the local well-posedness in the
space of analytic functions, and finite-time blowup of solution to the 3D inviscid PEs with rotation (Coriolis force). Moreover, I will also show, in the case
of “well-prepared” analytic initial data, the regularizing effect of the Coriolis force by providing a lower bound for the life-span of the solutions that grows
toward infinity with the rotation rate. This is a joint work with Quyuan Lin (Texas A&M) and Edriss S. Titi (Texas A&M and University of Cambridge).