Stability of the composite wave for compressible Navier-Stokes/Allen-Cahn system

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of two rarefaction waves and a viscous contact wave for the Cauchy problem to a one-dimensional compressible non-isentropic Navier-Stokes/Allen-Cahn system which is a combination of the classical Navier-Stokes system with an Allen-Cahn phase field description. We first construct the composite wave through Euler equations under the assumption of chi(x, t) 1 for the large time behavior, and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding Cauchy problem of the non-isentropic Navier-Stokes/Allen-Cahn system. The proof is mainly based on a basic energy method.