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Shape Optimization, Homogenization and Optimal Control pp 113–142Cite as

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Some Recent Developments in Optimal Control of Multiphase Flows

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 169))

Abstract

The present work serves as a review of some current developments in optimal control of two-phase flows. We discuss the upcoming analytical and numerical challenges and illustrate adequate solution strategies. This includes, among other things, the existence proof of solutions to a coupled Cahn–Hilliard–Navier–Stokes system, the derivation of first order optimality conditions for the associated optimal control problem in the presence of a nonsmooth free energy density. Moreover, we study spatial mesh adaptivity concepts based on a dual weighted residual approach and address future research directions concerned with the derivation of stronger stationarity conditions and/or the design of more efficient numerical solution algorithms.

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Acknowledgements

The authors gratefully acknowledge the support of the DFG through the priority program 1506 “Transport processes at fluidic interfaces” under the grant HI 1466/2-1 and the DFG-AIMS Workshop in Mbour, Sénégal. This research was further supported by the Research Center MATHEON through project C-SE5 and D-OT1 funded by the Einstein Center for Mathematics Berlin. In addition, this research was partly supported by the Berlin Mathematical School.

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Authors and Affiliations

  1. Weierstrass-Institut, Berlin, Germany

    Michael Hintermüller & Tobias Keil

  2. Humboldt-Universität zu Berlin, Institut für Mathematik, Berlin, Germany

    Michael Hintermüller & Tobias Keil

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  1. Michael Hintermüller
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Correspondence to Michael Hintermüller .

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Editors and Affiliations

  1. FB IV Abt. Mathematik, Universität Trier, Trier, Germany

    Volker Schulz

  2. Département de Mathématiques de la, Université Cheikh Anta Diop de Daka, Dakar-Fann, Senegal

    Diaraf Seck

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Hintermüller, M., Keil, T. (2018). Some Recent Developments in Optimal Control of Multiphase Flows. In: Schulz, V., Seck, D. (eds) Shape Optimization, Homogenization and Optimal Control . International Series of Numerical Mathematics, vol 169. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90469-6_7

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