Large Time Behavior of Solutions to Cauchy Problem for 1-D Compressible Isentropic Navier-Stokes/Allen-Cahn System

This paper is concerned with the large time behavior of the solutions to the Cauchy problem for the one-dimensional compressible Navier-Stokes/Allen-Cahn system with the immiscible two-phase flow initially located near the phase separation state. Under the assumptions that the initial data is a small perturbation of the constant state, we prove the global existence and uniqueness of the solutions and establish the time decay rates of the solution as well as its higher-order spatial derivatives. Moreover, we derive that the solutions of the system are time asymptotically approximated by the solutions of the modified parabolic system and obtain decay rates in $L^2$ and $L^1$. Furthermore, we show that the solution of the system is time asymptotically approximated in $L^p (1 \leq p \leq+\infty)$ by the diffusion waves.