Abstract
Abstract
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.
Funder
Deutsche Forschungsgemeinschaft
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Reference29 articles.
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2. Nonequilibrium thermodynamics and statistical physics of surfaces;Bedeaux,1986
3. Multivalued differential equations on graphs;Bothe;Nonlinear Anal.,1992
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