Large Time Behavior and Sharp Interface Limit of Compressible Navier-Stokes/Allen-Cahn System for Interacting Shock Waves

TL;DR

This paper analyzes the long-term behavior and sharp interface limit of a compressible Navier-Stokes/Allen-Cahn system modeling two-phase flow with shock interactions, establishing global solutions and convergence to entropy solutions.

Contribution

It provides the first rigorous analysis of the large time behavior and sharp interface limit for this coupled system with shock interactions.

Findings

Existence of a unique global solution near shock wave superpositions.

Asymptotic convergence to viscous shock and rarefaction waves.

Convergence to entropy solutions as interface thickness tends to zero.

Abstract

In this paper, we study the large time behavior and sharp interface limit of the Cauchy problem for compressible Navier-Stokes/Allen-Cahn system with interaction shock waves in the same family. This system is an important mathematical model for describing the motion of immiscible two-phase flow. The results show that, if the initial density and velocity are near the superposition of two shock waves in the same family, then there exists a unique global solution to the compressible Navier-Stokes/Allen-Cahn system, and this solution asymptotically converges to the superposition of the viscous shock wave and rarefaction wave which moving in opposite directions. Moreover, this global-in-time solution converges to the entropy solution of $p$-system in $L^\infty$-norm as the thickness of the diffusion interface tends to zero.