On a singular limit for the compressible rotating Euler system

TL;DR

This paper investigates the asymptotic behavior of a rotating compressible Euler system in low Mach and Rossby number regimes, identifying the quasi-geostrophic system as the limit and using dispersive estimates to handle oscillations.

Contribution

It introduces a novel analysis of the singular limit for the rotating compressible Euler system using dissipative measure-valued solutions and dispersive estimates.

Findings

The quasi-geostrophic system emerges as the limit in the low Mach and Rossby number regime.

Dispersive estimates effectively eliminate the impact of acoustic waves in the limit.

The approach handles ill-prepared initial data with rapidly oscillating acoustic waves.

Abstract

The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in case of ill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acoustic waves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in the asymptotic limit is eliminated.