Solvability of a~model problem for the Stokes equations in an infinite edge

V. A. Solonnikov

Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part 14, Tome 215 (1994), pp. 285-300

Résumé

The paper is concerned with a model problem for the Stokes equations on the half-plane $R^2_+\colon x_2>0$ with different boundary conditions on the half-axes ($x_2=0$, $x_10$) and ($x_2=0$, $x_1>0$). This model problem plays an important role in the investigation of some free boundary problems, such as problem of a filling or a drying a capillary. The proof of the solvability of the problem in weighted Sobolev and Hölder spaces is presented, estimates of the solution are obtained, the asymptotics formula for the solution in the neigbourhood of a singular point $x=0$ is derived. The proof is based on an explicit formula for the solution in terms of its Mellin transform, which makes it possible to obtain estimates uniform with respect to one of the parameters of the problem (in the problem of the filling a capillary it is proportional to the velocity of the filling). Bibliography: 9 titles.