Global weak solutions for a viscous liquid-gas model with singular pressure law
Global weak solutions for a viscous liquid-gas model with singular pressure law
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DOI：
10.3934/cpaa.2009.8.1867
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发表时间：
2009-08
影响因子：
1
通讯作者：
S. Evje;K. Karlsen
中科院分区：
文献类型：
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作者：
S. Evje;K. Karlsen
我们研究了一种适用于井和管流模拟的粘性气液两相模型。假设气体是多变的，而液体被视为不可压缩流体，导致压力定律，当发生向单相液体流动的转变时，该定律变得奇异。为了解决这一困难，我们用拉格朗日变量重新表述了模型，并在自由边界环境中研究了模型，其中气体和液体质量在初始时是紧密支撑的，并且在边界处是不连续的。然后，通过应用适当的变量变换，可以得到质量的逐点控制，从而保证当初始状态代表两相的真实混合物时，不会出现单相区。这为得到一类弱解的整体存在性结果铺平了道路。结果表明，粘性系数以适当的方式依赖于体积分数。通过假定初始流体速度更具正则性，得到了一类适当(较小)弱解的唯一性结果。
We study a viscous two-phase liquid-gas model relevant for well and pipe flow modelling. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid leading to a pressure law which becomes singular when transition to single-phase liquid flow occurs. In order to handle this difficulty we reformulate the model in terms of Lagrangian variables and study the model in a free-boundary setting where the gas and liquid mass are of compact support initially and discontinuous at the boundaries. Then, by applying an appropriate variable transformation, point-wise control on masses can be obtained which guarantees that no single-phase regions will occur when the initial state represents a true mixture of both phases. This paves the way for deriving a global existence result for a class of weak solutions. The result requires that the viscous coefficient depends on the volume fraction in an appropriate manner. By assuming more regularity of the initial fluid velocity a uniqueness result is obtained for an appropriate (smaller) class of weak solutions.