Wellposedness of the discontinuous ODE associated with two-phase flows

We consider the initial value problem $$\dot{x}(t)=v(t,x(t))fort\in (a,b),x(t_0)=x_0$$ which determines the pathlines of a two-phase flow, i.e.\ $v=v(t,x)$ is a given velocity field of the type $$v(t,x)=\{\begin{array}{ll}{v}^{+}(t,x)& \text{ if }x\in {\mathrm{\Omega }}^{+}(t)\\ v^{-}(t,x)& \text{ if }x\in {\mathrm{\Omega }}^{-}(t)\end{array}$$ with ${\mathrm{\Omega }}^{\pm }(t)$ denoting the bulk phases of the two-phase fluid system under consideration. The bulk phases are separated by a moving and deforming interface $\mathrm{\Sigma }(t)$. 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 $\mathrm{\Sigma }(t)$, 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$arecontinuousin$(t,x)$andlocallyLipschitzcontinuousin$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.