European Congress of Mathematics pp 263-272 | Cite as

# Evolution of a Closed Interface between Two Liquids of Different Types

## Abstract

We study a free boundary problem governing the motion of two immiscible viscous capillary fluids. The fluids occupy the whole space R^{3} 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.

## Keywords

Incompressible Fluid Plane Interface Free Boundary Problem Finite Time Interval Unique Solvability## Preview

Unable to display preview. Download preview PDF.

## References

- 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 - 2.I. V. Denisova,
*Evolution of compressible and incompressible fluids separated by a closed interface*, Interfaces and Free Boundaries 2 (2000), no. 3, 283–312.MathSciNetzbMATHCrossRefGoogle Scholar - 3.I. V. Denisova,
*Problem of the motion of two compressible fluids separated by a closed free interface*, Zap. Nauchn. Semin. Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 243 (1997), 61–86 (English transl. in J. Math. Sci. 99 (2000), no. 1, 837–853).MathSciNetGoogle Scholar - 4.I. V. Denisova,
*Problem of the motion of two viscous incompressible fluids separated by a closed free interface*, Acta Appl. Math. 37 (1994), 31–40.MathSciNetzbMATHGoogle Scholar - 5.I. V. Denisova and V. A. Solonnikov,
*Solvability of the linearized problem on the motion of a drop in a liquid flow*, Zap. Nauchn. Semin. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 171 (1989), 53–65 (English transi in J. Soviet Math.**56**(1991), no. 2, 2309–2316).MathSciNetGoogle 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 - 7.V. A. Solonnikov,
*Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval*,Algebra i Analiz 3 (1991), no. 1, 222–257 (English transl. in St.Petersburg Math. J. 3 (1992), no. 1, 189–220).MathSciNetGoogle 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