# Wellposedness of the discontinuous ODE associated with two-phase flows

## TL;DR

This paper establishes the existence and uniqueness of solutions for a discontinuous ODE modeling two-phase flows with phase change, ensuring well-defined flow paths in complex fluid systems.

## Contribution

It proves well-posedness of a discontinuous ODE for two-phase flows with phase change, under specific regularity conditions on velocity fields and interface transversality.

## Key findings

- Existence of solutions under continuous and Lipschitz velocity fields.
- Uniqueness of solutions with transversality condition at the interface.
- Foundation for modeling two-phase continua with phase change.

## Abstract

We consider the initial value problem \[ \dot x (t) = v(t,x(t)) \;\mbox{ for } t\in (a,b), \;\; x(t_0)=x_0 \] which determines the pathlines of a two-phase flow, i.e.\ $v=v(t,x)$ is a given velocity field of the type \[ v(t,x)= \begin{cases} v^+(t,x) &\text{ if } x \in \Omega^+(t)\\ v^-(t,x) &\text{ if } x \in \Omega^-(t) \end{cases} \] with $\Omega^\pm (t)$ denoting the bulk phases of the two-phase fluid system under consideration. The bulk phases are separated by a moving and deforming interface $\Sigma (t)$. Since we allow for flows with phase change, these pathlines are allowed to cross or touch the interface. Imposing a kind of transversality condition at $\Sigma (t)$, which is intimately related to the mass balance in such systems, we show existence and uniqueness of absolutely continuous solutions of the above ODE in case the one-sided velocity fields $v^\pm:\overline{{\rm gr}(\Omega^\pm)}\to \mathbb{R}^n$ are continuous in $(t,x)$ and locally Lipschitz continuous in $x$. Note that this is a necessary prerequisite for the existence of well-defined co-moving control volumes for two-phase flows, a basic concept for mathematical modeling of two-phase continua.