$\mathbf{BV}$ Solutions to $1$D Isentropic Euler Equations in the Zero Mach Number Limit

TL;DR

This paper rigorously analyzes the zero Mach number limit of coupled 1D compressible fluids, demonstrating convergence to incompressible behavior using a refined wave front tracking method.

Contribution

It introduces a novel wave front tracking algorithm that provides precise convergence estimates for the coupled fluid system as Mach number approaches zero.

Findings

Proves convergence of compressible to incompressible fluids in 1D

Develops a refined wave front tracking algorithm

Provides explicit convergence estimates dependent on Mach number

Abstract

Two compressible immiscible fluids in 1D and in the isentropic approximation are considered. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, we prove the rigorous convergence for the fully non--linear compressible to incompressible limit of the coupled dynamics of the two fluids. A key role is played by a suitably refined wave front tracking algorithm, which yields precise $\mathbf{BV}$, $\mathbf{L}^1$ and weak* convergence estimates, either uniform or explicitly dependent on the Mach number.