Uniqueness of the $1$D Compressible to Incompressible Limit

TL;DR

This paper studies the limit behavior of two immiscible fluids in one dimension as the Mach number approaches zero, establishing the uniqueness of the resulting coupled incompressible dynamics.

Contribution

It provides a characterization and proof of the uniqueness of the compressible to incompressible limit for two immiscible fluids in 1D.

Findings

The coupled dynamics converge to a unique limit as Mach number vanishes.

The limit satisfies a specific ODE--PDE system.

The result ensures well-posedness of the limit model.

Abstract

Consider two compressible immiscible fluids in 1D in the isentropic approximation. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, the coupled dynamics of the two fluids results as the compressible to incompressible limit and is known to satisfy an ODE--PDE system. Below, a characterization of this limit is provided, ensuring its uniqueness.