Solvability in Hölder spaces of a model initial-boundary value problem generated by a problem on the motion of two fluids

One studies the initial-boundary value problem for the Stokes' system, arising at the investigation of the nonstationary motion of two viscous fluids, separated by a free surface. Junction conditions are prescribed in the plane ×₃=0}. The consideration of the surface tension leads to a noncoercive integral term in the condition for the jump of the normal stresses. The unique solvability and estimates of the solution in Hölder classes of functions of the given model problem are proved with the aid of a theorem on Fourier multipliers and a significant part of the paper is devoted to the proof of the required modifications of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 188, pp. 5–44, 1991.