Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. We construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order $s>1$, and suggests that a suitable space for the well-posedness is Gevrey class of order $s=1$, which is exactly the space of analytic functions.

海洋和大气层的大规模动力学受原始方程式（PEs）支配。众所周知，三维粘性PE在全球范围内处于Sobolev空间中。另一方面，已知没有旋转的无粘性PE在Sobolev空间中不适，其平滑解可以在有限时间内形成奇异性。在本文中，我们在存在旋转的情况下扩展了上述结果。我们构造了旋转不粘PE的有限时间爆破解，并建立了旋转不粘PE在Sobolev空间中的不适定性，这是因为其围绕某一稳态背景流的扰动既是线性又是非线性不适性在Sobolev空间。它的线性不稳定性是开尔文-亥姆霍兹（Kelvin-Helmholtz）类型，类似于在涡旋片问题中出现的那种线性不稳定性。$s>\mathrm{1个}$，并指出适合摆正姿势的适当空间是Gevrey阶数 $s=\mathrm{1个}$，这恰恰是解析函数的空间。