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

TL;DR

This paper investigates the zero Mach number limit of compressible primitive equations with ill-prepared initial data, demonstrating convergence to incompressible primitive equations and analyzing oscillatory wave effects.

Contribution

It extends previous work by analyzing the zero Mach limit with ill-prepared data, accounting for high oscillating acoustic waves in complex domains.

Findings

Convergence to incompressible primitive equations established.

High oscillating acoustic waves are effectively managed.

Results apply to both $ ext{R}^2 imes 2 ext{T}$ and $ ext{T}^2 imes 2 ext{T}$ domains.

Abstract

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.