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Determining Modes, Nodes and Volume Elements for Stationary Solutions of the Navier-Stokes Problem Past a Three-Dimensional Body

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Abstract

In this paper we show that every solution of the three-dimensional exterior Navier-Stokes boundary-value problem, corresponding to a given non-zero, constant velocity at infinity (flow past a body) and belonging to a very general functional class, , can be determined by a finite number of parameters. Our results extend the analogous classical results by Foiaş & Temam [6, 7], and by Jones & Titi [14] for the interior problem. This extension is by no means trivial, in that all fundamental tools used in the case of the interior problem – such as compactness of the Sobolev embeddings, Poincaré's inequality, and the special basis constituted by eigenfunctions of the Stokes operator – are no longer available for the exterior problem. An important consequence of our results is that any solution in is uniquely determined by the knowledge of the associated velocity field only ``near'' the boundary. Just how ``near'' it has to be depends only on the Reynolds number and on the body.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Pittsburgh, 630 Benedum Hall, Pittsburgh, PA, 15261, U.S.A.

    Giovanni P. Galdi

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  1. Giovanni P. Galdi
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Correspondence to Giovanni P. Galdi.

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Communicated by V. Šverák

Dedicated to John Heywood on the occasion of his 65th birthday

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P. Galdi, G. Determining Modes, Nodes and Volume Elements for Stationary Solutions of the Navier-Stokes Problem Past a Three-Dimensional Body. Arch. Rational Mech. Anal. 180, 97–126 (2006). https://doi.org/10.1007/s00205-005-0395-0

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