On existence of solutions in plane quasistationary Stokes flow driven by surface tension

Recently, the free boundary problem of quasistationary Stokes flow of a mass of viscous liquid under the action of surface tension forces has been considered by R.W. HOPPER, L.K. ANTANOVSKII, and others. The solution of the Stokes equations is represented by analytic functions, and a time dependent. conformal mapping ont.o the flow domain is applied for the t.ransformation of the problem to the unit. disk. Two coupled Hilbert problems have t.o be solved there which leads to a Fredholm boundary integral equation. The solution of this equation determines the time evolution of the conformal mapping. The question of existence of a solution t.o this evolution problem for arbitrary (smooth) initial data has not. yet been answered completely. In this paper, local existence in time is proved using a theorem of OVSIANNII\OV on Cauchy problems in an appropriat.e scale of Banach spaces. The necessary estimates are obt.ained in a way t.hat is oriented at t.he a priori estimat.es for the solution given by ANTANOVSKII. In the case of small deviations from t.he st.ationary solution represented by a circle, these a priori estimates toget.her wit.h the local results are used to prove even global existence of the solution in t.ime.