Viscous Flows in Domains with a Multiply Connected Boundary
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 the general case, the solvability of this problem has been 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 a velocity vector in the case when the flow has an axis of symmetry and a plane of symmetry perpendicular to it, which also intersects each component of the boundary. Having available this estimate, we prove an existence theorem for the 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.
Mathematics Subject Classification (2000)35Q30 76N10 76D05
KeywordsNavier—Stokes equations incompressible liquid multiply connected boundary Dirichlet integral a priori estimate
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