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Large scale dynamics of the oceans and the atmosphere are governed by theprimitive equations (PEs). It is well-known that the three-dimensional viscousPEs is globally well-posed in Sobolev spaces. On the other hand, the inviscidPEs without rotation is known to be ill-posed in Sobolev spaces, and its smoothsolutions can form singularity in finite time. In this paper, we extend theabove results in the presence of rotation. First, we construct finite-timeblowup solutions to the inviscid PEs with rotation, and establish that theinviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that itsperturbation around a certain steady state background flow is both linearly andnonlinearly ill-posed in Sobolev spaces. Its linear instability is of theKelvin-Helmholtz type similar to the one appears in the context of vortexsheets problem. This implies that the inviscid PEs is also linearly ill-posedin Gevrey class of order $s > 1$, and suggests that a suitable space for thewell-posedness is Gevrey class of order $s = 1$, which is exactly the space ofanalytic functions.

Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid Primitive Equations with Rotation