一类大初始扰动下非等熵可压缩 Navier-Stokes-Allen-Cahn 方程静态解的稳定性——献给陈化教授 70 寿辰

摘要/Abstract

摘要：

该文主要研究半空间 $\mathbb{R}^+$ 中非等熵可压缩 Navier-Stokes-Allen-Cahn 方程外流问题静态解的时间渐近稳定性. 该模型可用于描述两种宏观上互不相溶的粘性可压缩流体混合物的运动. 当绝热指数 $\gamma$ 接近 1 时, 证明了一维非等熵可压缩 Navier-Stokes-Allen-Cahn 方程存在唯一的整体解, 且当时间 $t\rightarrow\infty$ 时, 该整体解收敛到非退化静态解. 该文要求流体的温度函数和相场变量的初始扰动, 以及静态解的强度都很小, 但是流体的密度和速度函数的初始扰动都可以任意大. 作者的分析基于基本的 $L^2$- 能量方法以及一些新技巧, 这些技巧充分考虑到了相场变量和静态解的影响.

关键词: 可压缩 Navier-Stokes-Allen-Cahn 方程, 外流问题, 非退化静态解的稳定性

Abstract:

This paper is mainly concerned with the time-asymptotic stability of stationary solutions to the outflow problem of the non-isentropic compressible Navier-Stokes-Allen-Cahn equations in the half space $\mathbb{R}^+$. The models can be used to describe the motion of a mixture of macroscopically immiscible two viscous compressible fluids. When the adiabatic exponent $\gamma$ is sufficiently close to $1$, we prove that the one-dimensional non-isentropic compressible Navier-Stokes-Allen-Cahn equations admits a unique global solution, which tends to the non-degenerate stationary solution as time goes to infinity. In this paper, the initial perturbations of temperature function and the phase field variable, and the strength of the stationary solution are required to be sufficiently small, but the initial perturbations of the density and velocity functions can be arbitrarily large. Our analysis is based on the basic $L^2$-energy method and some new techniques, which take into account the effect the phase field variable and the stationary solution.

Key words: compressible Navier-Stokes-Allen-Cahn equations, outflow problem, stability of the non-degenerate stationary solution

中图分类号:

O175.2

陈正争, 雷丹, 闫雨歆, 赵会江. 一类大初始扰动下非等熵可压缩 Navier-Stokes-Allen-Cahn 方程静态解的稳定性——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 452-472.

Zhengzheng Chen, Dan Lei, Yuxin Yan, Huijiang Zhao. Stability of Stationary Solutions to the Non-Isentropic Compressible Navier-Stokes-Allen-Cahn Equations Under a Class of Large Initial Data[J]. Acta mathematica scientia,Series A, 2026, 46(2): 452-472.

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