Global Solution for Gas-Liquid Flow of 1-D van der Waals Equation of State with Large Initial Data

TL;DR

This paper proves the global existence and uniqueness of strong solutions for a 1-D diffuse interface model of gas-liquid phase transition using van der Waals equation, ensuring physical bounds for large initial data.

Contribution

It introduces new techniques to establish uniform density bounds and handle non-convex pressure functions in a 1-D van der Waals-based phase transition model.

Findings

Global strong solutions exist for large initial data.

Density and phase variables remain within physical bounds.

The model's solutions are unique and globally well-posed.

Abstract

This paper is concerned with a diffuse interface model for the gas-liquid phase transition. The model consists the compressible Navier-Stokes equations with van der Waals equation of state and a modified Allen-Cahn equation. The global existence and uniqueness of strong solution with the periodic boundary condition (or the mixed boundary condition) in one dimensional space is proved for large initial data. Furthermore, the phase variable and the density of the gas-liquid mixture are proved to stay in the physical reasonable interval. The proofs are based on the elementary energy method and the maximum principle, but with new development, where some techniques are introduced to establish the uniform bounds of the density and to treat the non-convexity of the pressure function.