On moving hypersurfaces and the discontinuous ODE-system associated with two-phase flows

We consider the initial value problem x ̇ ( t ) = v ( t , x ( t ) ) for   t ∈ ( 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 ) = v + ( t , x ) if x ∈ Ω + ( t ) v − ( t , x ) if x ∈ Ω − ( t ) with Ω ± ( t ) denoting the bulk phases of the two-phase fluid system under consideration. The bulk phases are separated by a moving and deforming interface Σ( t ) at which v can have jump discontinuities. Since flows with phase change are included, the pathlines are allowed to cross or touch the interface. Imposing a kind of transversality condition at Σ( 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 ± are continuous in ( t , x ) and locally Lipschitz continuous in x on their respective domain of definition. A main step in proving this result, also interesting in itself, is to freeze the interface movement by means of a particular coordinate transform which requires a tailor-made extension of the intrinsic velocity field underlying a C 1 , 2 -family of moving hypersurfaces.