On singular limits arising in the scale analysis of stratified fluid flows

TL;DR

This paper investigates the singular limits of stratified fluid flows in the low Mach and low Froude number regime, demonstrating convergence of solutions to the anelastic system under various initial data conditions.

Contribution

It provides a rigorous analysis of the convergence of compressible Navier-Stokes solutions to the anelastic system in stratified flows, covering both well-prepared and ill-prepared initial data.

Findings

Weak solutions converge to the anelastic Navier-Stokes system

Results hold for well-prepared initial data on flat domains

Convergence also established for ill-prepared data on infinite slabs

Abstract

We study the low Mach low Freude numbers limit in the compressible Navier-Stokes equations and the transport equation for evolution of an entropy variable -- the potential temperature $\Theta$. We consider the case of well-prepared initial data on "flat" tours and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions to the primitive system converge to the solution to the anelastic Navier-Stokes system and the transport equation for the second order variation of $\Theta$