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

TL;DR

This paper investigates the long-term behavior of solutions to a one-dimensional compressible Navier-Stokes/Allen-Cahn system, proving global existence, decay rates, and asymptotic approximations for small initial perturbations.

Contribution

It establishes the global existence, uniqueness, and decay rates of solutions, and shows their asymptotic approximation by diffusion waves and modified parabolic systems.

Findings

Solutions exist globally and are unique for small initial perturbations.

Solutions decay over time with established rates in various norms.

Solutions are asymptotically approximated by diffusion waves and modified systems.

Abstract

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.