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

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Abstract

We consider a model problem for the Stokes equations in the half-plane ℝ 2+ (x2>0) with different boundary conditions on the semiaxes (x2=0, x1<0) and (x2=0, x1>0), which plays an important role in the studies of some free boundary problems, such as problem of filling or drying a capillary. The proof of the solvability of the problem in weighted Sobolev and Hölder spaces is presented, and estimates for the solution as well as the asymptotic formula for the solution in the vicinity of the singular point x=0 are obtained. The proof is based on an explicit formula for the solution in terms of its Mellin transform, which makes it possible to obtain the estimates uniform with respect to one of the parameters of the problem (in the problem of filling a capillary it is proportional to the velocity of filling). Bibliography: 9 titles.

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Authors
  1. V. A. Solonnikov
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Dedicated to L. D. Faddeev on the occasion of his 60th birthday

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 285–300.

Translated by V. A. Solonnikov.

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Solonnikov, V.A. Solvability of a model problem for the Stokes equations in an infinite angle. J Math Sci 85, 1741–1751 (1997). https://doi.org/10.1007/BF02355335

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  • DOI: https://doi.org/10.1007/BF02355335

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