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We study the effect of the rotation on the life-span of solutions to the $3D$hydrostatic Euler equations with rotation and the inviscid Primitive equations(PEs) on the torus. The space of analytic functions appears to be the naturalspace to study the initial value problem for the inviscid PEs with generalinitial data, as they have been recently shown to exhibit Kelvin-Helmholtz typeinstability. First, for a short interval of time that is independent of therate of rotation $|\Omega|$, we establish the local well-posedness of theinviscid PEs in the space of analytic functions. In addition, thanks to a fineanalysis of the barotropic and baroclinic modes decomposition, we establish tworesults about the long time existence of solutions. (i) Independently of$|\Omega|$, we show that the life-span of the solution tends to infinity as theanalytic norm of the initial baroclinic mode goes to zero. Moreover, we show inthis case that the solution of the $3D$ inviscid PEs converges to the solutionof the limit system, which is governed by the $2D$ Euler equations. (ii) Weshow that the life-span of the solution can be prolonged unboundedly with$|\Omega|\rightarrow \infty$, which is the main result of this paper. This isestablished for "well-prepared" initial data, namely, when only the Sobolevnorm (but not the analytic norm) of the baroclinic mode is small enough,depending on $|\Omega|$. Furthermore, for large $|\Omega|$ and "well-prepared"initial data, we show that the solution to the $3D$ inviscid PEs isapproximated by the solution to a simple limit resonant system with the sameinitial data.

On the effect of rotation on the life-span of analytic solutions to the $3D$ inviscid primitive equations