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Shape Optimization for Navier-Stokes Equations

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Control of Coupled Partial Differential Equations

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 155))

Abstract

The minimization of drag functional for the stationary, isothermal, compressible Navier-Stokes equations (N-S-E) in three spatial dimensions is considered. In order to establish the existence of an optimal shape the general result [26] on compactness of families of generalized solutions to N-S-E is applied. The family of generalized solutions to N-S-E is constructed over a family of admissible domains \( \mathcal{U}_{ad} \). Any admissible domain Ω = B\S contains an obstacle S, e.g., a wing profile. Compactness properties of the family of admissible domains are imposed. It turns out that we require the compactness of the family of admissible domains with respect to the Hausdorff metrics as well as in the sense of Kuratowski-Mosco. The analysis is performed for the range of adiabatic ratio γ > 1 in the pressure law p(ρ) = gr γ and it is based on the technique proposed in [24] for the discretized N-S-E.

This work was completed during a stay of the first author at the Institute Elie Cartan of the University Henri Poincaré Nancy I.

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Author information

Authors and Affiliations

  1. Siberian Division of Russian Academy of Sciences, Lavryentyev Institute of Hydrodynamics, Lavryentyev pr. 15, Novosibirsk, 630090, Russia

    Pavel I. Plotnikov

  2. Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, F-54506, Vandoeuvre lès Nancy Cedex, France

    Jan Sokolowski

Authors
  1. Pavel I. Plotnikov
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  2. Jan Sokolowski
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Editor information

Editors and Affiliations

  1. Institute for Mathematics, University of Graz, Heinrichstraße 36, A-8010, Graz, Austria

    Karl Kunisch

  2. Institute for Mathematics, Humboldt University Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    Jürgen Sprekels

  3. Institute for Applied Mathematics, University of Erlangen-Nürnberg, Martensstraße 3, D-91058, Erlangen, Germany

    Günter Leugering

  4. Institute for Mathematics, Technical University Berlin, Straße des 17 Juni 136, D-10623, Berlin, Germany

    Fredi Tröltzsch

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© 2007 Birkhäuser Verlag Basel/Switzerland

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Plotnikov, P.I., Sokolowski, J. (2007). Shape Optimization for Navier-Stokes Equations. In: Kunisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (eds) Control of Coupled Partial Differential Equations. International Series of Numerical Mathematics, vol 155. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7721-2_11

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