Dispersion of compressible rotating Euler equations with low Mach and Rossby numbers

In this paper, we consider the low Mach and Rossby number singular limits and longtime existence of strong solution to the initial value problem of 3D compressible rotating Euler equations with ill-prepared initial data. We establish the Strichartz decay estimates that are uniform to the Mach number, the Rossby number, and the ratio of these two parameters for the associated linear propagator without any restrictions on the frequency. In particular, difficulties arisen from the degeneracy of the phase function and the vanishing of the ratio of the two parameters are addressed by elaborately designed splitting techniques and discussions for each frequencies. Using the decay estimates, we prove the longtime existence and obtain a rate of convergence to zero of strong solution to the compressible rotating Euler equations with initial data of finite energy in $\mathbb{R}^3$.