Uniqueness of the $1$D Compressible to Incompressible Limit

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.