We consider the initial value problem ##IMG## [http://ej.iop.org/images/0951-7715/33/10/5425/nonab987dieqn1.gif] {$\dot {x}\left(t\right)=v\left(t,x\left(t\right)\right)\;\text{for}\,\;t\in \left(a,b\right),x\left({t}_{0}\right)={x}_{0}$} which determines the pathlines of a two-phase flow, i.e. v = v ( t , x ) is a given velocity field of the type ##IMG## [http://ej.iop.org/images/0951-7715/33/10/5425/nonab987dieqn2.gif] {$v\left(t,x\right)=\begin{cases}^{+}\left(t,x\right)\;\text{if}\;x\in {{\Omega}}^{+}\left(t\right)\hfill \\ {v}^{-}\left(t,x\right)\;\text{if}\;x\in {{\Omega}}^{-}\left(t\right)\hfill \end{cases}$} 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...

中文翻译：

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关于运动的超曲面和与两相流相关的不连续ODE系统

我们考虑初始值问题## IMG ## [http://ej.iop.org/images/0951-7715/33/10/5425/nonab987dieqn1.gif] {$ \ dot {x} \ left（t \ right）= v \ left（t，x \ left（t \ right）\ right）\; \ text {for} \，\; t \ in \ left（a，b \ right），x \ left（{t } _ {0} \ right）= {x} _ {0} $}，它确定两相流的路径，即v = v（t，x）是类型为## IMG＃的给定速度场＃[http://ej.iop.org/images/0951-7715/33/10/5425/nonab987dieqn2.gif] {$ v \ left（t，x \ right）= \ begin {cases} ^ {+} \ left（t，x \ right）\; \ text {if} \; x \ in {{\ Omega}} ^ {+} \ left（t \ right）\ hfill \\ {v} ^ {-} \ left（t，x \ right）\; \ text {if} \; x \ in {{\ Omega}} ^ {-} \ left（t \ right）\ height \ end {cases $$}用Ω±（ t）表示所考虑的两相流体系统的整体相。主体相由运动和变形界面Σ（t）隔开，在该界面处v可能具有跳跃不连续性。