Global existence of solutions for a multi-phase flow: a bubble in a liquid tube and related cases

TL;DR

This paper proves the global existence of weak solutions for a three-phase flow system modeling phase transitions in fluids, under certain stability and initial data conditions, using a front tracking algorithm.

Contribution

It establishes the global existence of solutions for a multi-phase flow model with large initial data, a significant extension in the mathematical theory of phase transition systems.

Findings

Global solutions exist for initial data with BV-norm below a large threshold.

The system models phase transitions with stationary interfaces in a one-dimensional setting.

A front tracking algorithm is used to prove the existence of solutions.

Abstract

In this paper we study the problem of the global existence (in time) of weak, entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit (large) threshold, then the Cauchy problem has global solutions.