可压缩非理想非混相两相流的一维初值问题

Initial Value Problem for Compressible Non-Ideal Immiscible Two-Phase Flow in 1D

摘要: 可压缩Navier-Stokes/Allen-Cahn方程组是描述非混相两相流流动的重要数学模型之一。本文研究可压缩等熵(或等温) Navier-Stokes/Allen-Cahn方程组初值问题的一维全局强解的存在唯一性，其中压强满足Van der Waals状态方程，该状态方程关于密度是非单调的，物理上常用来描述流体的相变。本文通过能量方法，在初值不含真空的情况下，证明了该方程组一维Cauchy问题全局强解的存在唯一性，这一结论不需要对初值加以任何小性限制。结果表明，在任意有限时间段内，非混相两相流的密度、相场、速度等物理量保持连续性，在可能的相变发生处，流体的密度可以振荡剧烈，但是不会出现间断，新的相变产生之处，不同的相之间由扩散界面连接。

Abstract: The compressible Navier-Stokes/Allen-Cahn equations system is one of the important mathematical models for describing immiscible two-phase flows. This paper studies the existence and uniqueness of global strong solutions to the initial value problem for the 1-D compressible isentropic (or isothermal) Navier-Stokes/Allen-Cahn equations. The pressure satisfies the Van der Waals equation of state, which is non-monotonic with respect to density and is commonly used in physics to describe fluid phase transitions. Using the energy method, we establish the existence and uniqueness of global strong solutions for the equations under the condition that the initial data contains no vacuum. This result holds without imposing any smallness restriction on the initial data. The results demonstrate that within any finite time interval, the physical quantities of the immiscible two-phase flow—such as density, phase field, and velocity—remain continuous. While the density may exhibit intense oscillations at locations where phase transitions are possible, discontinuities do not arise. Furthermore, where new phases emerge, the distinct phases are connected by diffuse interfaces.