Phase Transition in Non-isentropic Compressible Immiscible Two-Phase Flow with van der Waals Equation of State

TL;DR

This paper proves the existence and uniqueness of global solutions for a complex two-phase flow model governed by the van der Waals equation, showing that phase transitions do not lead to unbounded physical quantities over finite times.

Contribution

It establishes the global well-posedness of a non-isentropic Navier-Stokes/Allen-Cahn system with van der Waals EOS, including phase transitions, without small initial data assumptions.

Findings

Global strong solutions exist and are unique.

Physical quantities remain bounded despite phase transitions.

No smallness restrictions on initial conditions.

Abstract

This study establishes the global well-posedness of the compressible non-isentropic Navier-Stokes/Allen-Cahn system governed by the van der Waals equation of state $p(\rho,\theta)=- a\rho^2+\frac{R\theta\rho}{1-b\rho}$ and degenerate thermal conductivity $\kappa(\theta)=\tilde{\kappa}\theta^\beta$, where $p$, $\rho$ and $\theta$ are the pressure, the density and the temperature of the flow respectively, and $a,b,R,\tilde\kappa$ are positive constants related to the physical properties of the flow. Navier-Stokes/Allen-Cahn system models immiscible two-phase flow with diffusive interfaces, where the non-monotonic pressure-density relationship in the van der Waals equation drives gas-liquid phase transitions. By developing a refined $L^2$-energy framework, we prove the existence and uniqueness of global strong solutions to the one-dimensional Cauchy problem for non-vacuum and finite-temperature initial data, without imposing smallness restrictions on the initial conditions. The findings demonstrate that despite non-monotonic pressure inducing substantial density fluctuations and triggering phase transitions, all physical quantities remain bounded over finite time intervals.