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• W3083122412 abstract "Abstract We consider the initial value problem <?CDATA $dot {x}left(tright)=vleft(t,xleft(tright)right);text{for},;tin left(a,bright),xleft({t}_{0}right)={x}_{0}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mrow> <mml:mover accent=true> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>̇</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>v</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mspace width=thinmathspace /> <mml:mtext>for</mml:mtext> <mml:mo> </mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> which determines the pathlines of a two-phase flow, i.e. v = v ( t , x ) is a given velocity field of the type <?CDATA $vleft(t,xright)=begin{cases}^{+}left(t,xright);text{if};xin {{Omega}}^{+}left(tright)hfill {v}^{-}left(t,xright);text{if};xin {{Omega}}^{-}left(tright)hfill end{cases}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>v</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfenced close= open={> <mml:mrow> <mml:mtable class=cases columnspacing=1> <mml:mtr> <mml:mtd columnalign=left> <mml:msup> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mspace width=thinmathspace /> <mml:mtext>if</mml:mtext> <mml:mspace width=thinmathspace /> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=normal>Ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd columnalign=left> <mml:msup> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mspace width=thinmathspace /> <mml:mtext>if</mml:mtext> <mml:mspace width=thinmathspace /> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=normal>Ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mfenced> </mml:math> 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 <?CDATA ${mathcal{C}}^{1,2}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:msup> <mml:mrow> <mml:mi mathvariant=script>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> -family of moving hypersurfaces." @default.
• W3083122412 created "2020-09-11" @default.
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• W3083122412 date "2020-09-02" @default.
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• W3083122412 title "On moving hypersurfaces and the discontinuous ODE-system associated with two-phase flows" @default.
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