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Note on the flux condition and pressure drop in the resolvent problem of the Stokes system

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Abstract

In [4] H. Sohr and the author considered theL q-theory of the resolvent problem of the generalized Stokes system in an aperture domain. This type of unbounded domain consists of two disjoint half spaces which are separated from each other by a wall but connected by a hole (aperture) in this wall. Due to this geometry the flux of the velocity field through the hole and the pressure drop at infinity are important physical and mathematical quantities. In this note we show that in order to single out a unique solution of the resolvent problem we must prescribe the flux for largeq, but that for smallq neither the flux nor the pressure drop can be prescribed. Only if the dimension is greater than two there is a certain range of values ofq where we must prescribe either the flux or the pressure drop. As a limit case we also investigate strong solutions of the Stokes system.

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  1. Fachbereich Mathematik, Technische Hochschule Darmstadt, Schloßgartenstr. 7, D-64289, Darmstadt

    Reinhard Farwig

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  1. Reinhard Farwig
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Farwig, R. Note on the flux condition and pressure drop in the resolvent problem of the Stokes system. Manuscripta Math 89, 139–158 (1996). https://doi.org/10.1007/BF02567510

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  • DOI: https://doi.org/10.1007/BF02567510

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