Stability of the Rarefaction Wave for a Full Compressible Navier-Stokes/Allen-Cahn Equations

Abstract

This paper is concerned with the non-isentropic compressible Navier-Stokes/Allen-Cahn equations with the diffusion interface, which is an important mathematical model in the numerical simulation of compressible immiscible two-phase flow. When the space-asymptotic states (v_±, u_±, θ_±) lie in the rarefaction curve of the Riemann problem of the compressible Euler equations, we prove that the time-asymptotic state of solutions to the 1-D Cauchy problem is the rarefaction wave, that is, the stability of the rarefaction wave, where the strength of the rarefaction wave is not required to be small. Moreover, we consider the general gases including ideal polytropic gas and allow the different space-asymptotic states χ_± for the concentration difference of the mixture fluids. The proof is mainly based on a basic energy method. By product, we give the proof of the uniqueness of the global solutions to the 1-D Cauchy problem.