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On generalized solutions of two-phase flows for viscous incompressible fluids. (English) Zbl 1124.35060
The existence of generalized solutions of the non-stationary flow of two immiscible, incompressible, viscous Newtonian and non-Newtonian fluids with and without surface tension in a domain $$\Omega \subseteq {\mathbb R}^{d}, d=2,3,$$ is investigated. In the case without surface tension the existence of weak solutions can be easier shown. But there is no knowledge about the interface between both fluids. The author proves the existence of so-called measure-valued varifold solutions, where the interface is modeled by an oriented general varifold $$V(t)$$ which is a non-negative measure on $$\Omega \times {\mathbb S}^{d-1}$$, where $${\mathbb S}^{d-1}$$ is the unit sphere in $${\mathbb R}^{d}$$. Furthermore, it is shown that measure-valued varifold solutions are weak solutions if an energy equality is satisfied.

MSC:
 35Q35 PDEs in connection with fluid mechanics 76D27 Other free boundary flows; Hele-Shaw flows 76D45 Capillarity (surface tension) for incompressible viscous fluids 76T99 Multiphase and multicomponent flows
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