Zero Mach Number Limit of the Compressible Primitive Equations: Ill-prepared Initial Data

In the work, we consider the zero Mach number limit of compressible primitive equations in the domain ${\mathbb{R}}^2\times 2\mathbb{T}$ or ${\mathbb{T}}^2\times 2\mathbb{T}$. We identify the limit equations to be the primitive equations with the incompressible condition. The convergence behaviors are studied in both ${\mathbb{R}}^2\times 2\mathbb{T}$ and ${\mathbb{T}}^2\times 2\mathbb{T}$, respectively. This paper takes into account the high oscillating acoustic waves and is an extension of our previous work by X. Liu and E.S. Titi, Arch. Rational Mech. Anal., 238, 705-747, 2020.