Solvability of a linearized problem on a motion of a drop in a fluid flow

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.