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November  2009, 8(6): 1867-1894. doi: 10.3934/cpaa.2009.8.1867

Global weak solutions for a viscous liquid-gas model with singular pressure law

Steinar Evje 1, and  Kenneth Hvistendahl Karlsen 2,

1. 

Centre of Mathematics for Applications (CMA), University of Oslo, 1053 Blindern, NO-0316 Oslo, Norway
2. 

Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO-0316 Oslo, Norway

Received  November 2008 Revised  March 2009 Published  August 2009

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.
Keywords: Two-phase flow, Lagrangian coordinates, free boundary problem., weak solutions.
Mathematics Subject Classification: Primary: 76T10, 76N10, 65M12; Secondary: 35L6.
Citation: Steinar Evje, Kenneth Hvistendahl Karlsen. Global weak solutions for a viscous liquid-gas model with singular pressure law. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1867-1894. doi: 10.3934/cpaa.2009.8.1867
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