Abstract
We study a free boundary problem governing the motion of two immiscible viscous capillary fluids. The fluids occupy the whole space R3 but one of them should have a finite volume. Every liquid may be of both types: compressible and incompressible.
Local (with respect to time) unique solvability of the problem is obtained in the Sobolev-Slobodetskii spaces. After the passage to Lagrangian coordinates, one obtains a nonlinear, noncoercive initial boundary-value problem the proof of the existence theorem for which is based on the method of successive approximations and on an explicit solution of a model linear problem with a plane interface between the liquids.
Some restrictions to the fluid viscosities appear in the case when at least, one of the liquids is compressible.
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Denisova, I.V. (2001). Evolution of a Closed Interface between Two Liquids of Different Types. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_22
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DOI: https://doi.org/10.1007/978-3-0348-8266-8_22
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