Sharp Interface Limit for Compressible Immiscible Two-Phase Dynamics with Relaxation

In this paper, the sharp interface limit for compressible Navier-Stokes/Allen-Cahn system with relaxation is investigated, which is motivated by the Jin-Xin relaxation scheme ([Comm.Pure Appl.Math.,48,1995]). Given any entropy solution which consists of two different families of shocks interacting at some positive time for the immiscible two-phase compressible Euler equations, it is proved that such entropy solution is the singular limit for a family global strong solutions of the compressible Navier-Stokes/Allen-Cahn system with relaxation when the interface thickness of immiscible two-phase flow tends to zero. The weighted estimation and improved anti-derivative method are used in the proof. The results of this singular limit show that, the sharp interface limit of the compressible Navier-Stokes/Allen-Cahn system with relaxation is the immiscible two-phase compressible Euler equations with free interface between phases. Moreover, the interaction of shock waves belong to different families can pass through the two-phase flow interface and maintain the wave strength and wave speed without being affected by the interface for immiscible compressible two-phase flow.