Geodesic
Solvability of a linearized problem on a motion of a drop in a fluid flow
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 20, Tome 171 (1989), pp. 53-65

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We prove the solvability of a linear problem which is generated by a problem on an unsteady motion of a drop in a vicons flow. We take into account a surface tension which enters in the boundary conditions for a jump of normal stresses as a non-coersive term containing the integral with respect to $t$. The vector field of velocities needs not be solenoidal but its divergence should be of a special form. The proof of the solvability is carried out in the spaces of Sobolev–Slobodetski, and it relies on a-priori estinates for solutions of the problem. 
@article{ZNSL_1989_171_a3,
     author = {I. V. Denisova and V. A. Solonnikov},
     title = {Solvability of a linearized problem on a motion of a drop in a fluid flow},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {53--65},
     publisher = {mathdoc},
     volume = {171},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1989_171_a3/}
}
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I. V. Denisova; V. A. Solonnikov. Solvability of a linearized problem on a motion of a drop in a fluid flow. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 20, Tome 171 (1989), pp. 53-65. http://geodesic.mathdoc.fr/item/ZNSL_1989_171_a3/