A coupling between a non-linear 1D compressible-incompressible limit and the 1D p-system in the non smooth case

We consider two compressible immiscible fluids in one space dimension and in the isentropic approximation. The first fluid is surrounded and in contact with the second one. As the sound speed of the first fluid diverges to infinity, we present the proof of rigorous convergence for the fully non-linear compressible to incompressible limit of the coupled dynamics of the two fluids. A linear example is considered in detail, where fully explicit computations are possible.