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

Large scale dynamics of the oceans and the atmosphere are governed by the\nprimitive equations (PEs). It is well-known that the three-dimensional viscous\nPEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid\nPEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth\nsolutions can form singularity in finite time. In this paper, we extend the\nabove results in the presence of rotation. First, we construct finite-time\nblowup solutions to the inviscid PEs with rotation, and establish that the\ninviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its\nperturbation around a certain steady state background flow is both linearly and\nnonlinearly ill-posed in Sobolev spaces. Its linear instability is of the\nKelvin-Helmholtz type similar to the one appears in the context of vortex\nsheets problem. This implies that the inviscid PEs is also linearly ill-posed\nin Gevrey class of order $s > 1$, and suggests that a suitable space for the\nwell-posedness is Gevrey class of order $s = 1$, which is exactly the space of\nanalytic functions.\n