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Title: Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential
Authors: Frigeri, SergioGrasselli, MaurizioSprekels, Jürgen
Publishers Version: https://doi.org/10.20347/WIAS.PREPRINT.2473
Issue Date: 2018
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 2473, ISSN 2198-5855
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract: In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the NavierStokes equations, nonlinearly coupled with a convective nonlocal CahnHilliard equation. The system rules the evolution of the (volume-averaged) velocity u of the mixture and the (relative) concentration difference ' of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force v acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map v 7! [u; '], and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with E. Rocca in [19]. There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and C. G. Gal in [14].
Keywords: Navier–Stokes equations; nonlocal Cahn–Hilliard equations; degenerate mobility; incompressible binary fluids; phase separation; distributed optimal control; first-order necessary optimality conditions
DDC: 510
License: This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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