Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Harald.Garcke@mathematik.uni-regensburg.de; Claudia.Hecht@mathematik.uni-regensburg.de

Received: 21 July 2014
Revised: 2 February 2015

Abstract

We apply a phase field approach for a general shape optimization problem of a stationary Navier-Stokes flow. To be precise we add a multiple of the Ginzburg–Landau energy as a regularization to the objective functional and relax the non-permeability of the medium outside the fluid region. The resulting diffuse interface problem can be shown to be well-posed and optimality conditions are derived. We state suitable assumptions on the problem in order to derive a sharp interface limit for the minimizers and the optimality conditions. Additionally, we can derive a necessary optimality system for the sharp interface problem by geometric variations without stating additional regularity assumptions on the minimizing set.