Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids
Helmut Abels and Matthias Röger
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Submission date: 21. Oct. 2008
published in: Annales de l'Institut Henri Poincaré / C, 26 (2009) 6, p. 2403-2424
DOI number (of the published article): 10.1016/j.anihpc.2009.06.002
MSC-Numbers: 35R35, 35Q30, 76D05, 76T99, 80A20
Keywords and phrases: two-phase flow, navier-stokes, free boundary problems, mullins-sekerka
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We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier--Stokes and Mullins--Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.