Evolution of a Closed Interface between Two Liquids of Different Types
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.
KeywordsIncompressible Fluid Plane Interface Free Boundary Problem Finite Time Interval Unique Solvability
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- 1.I. V. Denisova, A priori estimates of the solution of a linear time-dependent problem connected with the motion of a drop in a fluid medium, Trudy Mat. Inst. Steklov 188 (1990), 3–21 (English transl. in Proc. Steklov Inst. Math. (1991), no. 3, 1–24).Google Scholar
- 6.V. A. Solonnikov, On an initial-boundary value problem for the Stokes systems arising in the study of a problem with a free boundary, Trudy Mat. Inst. Steklov. 188 (1990), 150–188 (English transi. in Proc. Steklov Inst. Math. (1991), no. 3, 191–239).Google Scholar
- 8.V. A. Solonnikov and A. Tani, Free boundary problem for a viscous compressible flow with surface tension, in: Constantin Carathéodory: An International Tribute, World Scientific (1991), 1270–1303.Google Scholar