Viscous Flows in Domains with a Multiply Connected Boundary
Vladislav V. Pukhnachev
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Submission date: 25. Sep. 2008 (revised version: January 2009)
published in: New directions in mathematical fluid mechanics : the Alexander V. Kazhikhov memorial volume / A. V. Fursikov ... (eds.)
Basel : Birkhäuser, 2010. - P. 333 - 348
(Advances in mathematical fluid mechanics)
DOI number (of the published article): 10.1007/978-3-0346-0152-8_17
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In this paper we consider stationary Navier-Stokes equations in a bounded domain with a boundary, which has several connected components. The velocity vector is given on the boundary, where the fluxes differ from zero on its components. In general case, the solvability of this problem is an open question up to now. We provide a survey of previous results, which deal with partial versions of the problem. We construct an a priori estimate of the Dirichlet integral for velocity vector in the case, when the flow has two mutually perpendicular planes of symmetry, and, moreover, the line of their intersection intersects each component of the boundary. Having available this estimate, we prove the existence theorem for axially symmetric problem in a domain with a multiply connected boundary. We consider also the problem in a curvilinear ring and formulate a conditional result concerning its solvability.