Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow (Q2808046)

The authors deal with a shape optimization problem of a stationary Navier-Stokes flow of the form : NEWLINE\[NEWLINE \min_{(\varphi,\mathbf{u})}\;J_\varepsilon(\varphi,\mathbf{u}):= \frac 12 \int_\varOmega a_\varepsilon(\varphi)|\mathbf{u}|^2\,dx + \gamma\int_\varOmega \left[\frac \varepsilon 2|\nabla \varphi|^2+\frac 1\varepsilon\psi(\varphi)\right]\,dx,NEWLINE\]NEWLINE subject to \(\;(\varphi,\mathbf{u})\in \varPhi_{ad}\times U\;\) and NEWLINE\[NEWLINE\int_\varOmega \left[\alpha_\varepsilon(\varphi)\mathbf{u}\cdot \mathbf{v}+\mu \nabla \mathbf{u}\cdot \nabla\mathbf{v}\right]\,dx +b(\mathbf{u},\mathbf{u},\mathbf{v})=\int_\varOmega \mathbf{f}\cdot \mathbf{v}\,dx\;\forall\,\mathbf{v}\in \mathbf{V}, NEWLINE\]NEWLINE where \(\mathbf{V}:= \{\mathbf{v}\in \mathbf{H^1_0(\varOmega)\,|\,\text{div}\,\mathbf{v}=0}\}\). and the design space has the form NEWLINE\[NEWLINE\varPhi:= \left\{\varphi\in H^1(\varOmega)\,|\,|\varphi|\leq 1\,\text{a.e. in}\,\varOmega,\,\int_\varOmega \varphi\,dx\leq \beta| \varOmega| \right\},\;\beta\in (-1,1).NEWLINE\]NEWLINE The existence of at least one minimizer is verified and optimality conditions on the diffuse interface setting are derived. The sharp interface limit for the minimizers and the optimality conditions are derived under suitable assumptions. A necessary optimality system for the sharp interface problem are derived by geometric variations without stating additional regularity assumptions on the minimizing set.