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Zbl 0889.49027
Dziri, Raja; Zolésio, Jean-Paul
Shape existence in Navier-Stokes flow with heat convection. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No.1, 165-192 (1997). ISSN 0391-173X

The paper deals with a shape optimization problem having as state equation a stationary Navier-Stokes equation $$\cases -\kappa_1\Delta u+Du\cdot u+\nabla p & =f_1(y)\quad \text{in }\Omega\\ -\kappa_2\Delta u+Du\cdot u+\nabla q & =f_2(y)\quad \text{in }D\setminus\Omega\\ \text{ div} u & =0\quad\text{in }D\endcases$$ where $u$ is the velocity field, coupled with a heat equation for the temperature $y$. Across the interface $\Gamma=D\cap\partial\Omega$ we have the boundary conditions $u\cdot n=0$ and $\partial y/\partial n=0$. The cost $e_\theta(\Omega)$ has the form $$ \int_D\left(\textstyle{\frac12}|u|^2+\delta_1 \kappa_\Omega|\varepsilon (u)|^2+\delta_2{\mu_\Omega\over 2}|\nabla y|^2+ \rho g x_3\right) dx +\theta P_D \Omega $$ where $\rho$ is the density, $\delta_1, \delta_2, \theta>0$, $$\kappa_\Omega=\cases\kappa_1\text{ in }\Omega\\ \kappa_2\text{ in }D\setminus\Omega\endcases,\quad\mu_\Omega=\cases\mu_1\text{ in }\Omega\\ \mu_2\text{ in }D\setminus\Omega\endcases,$$ and $P_D(\Omega)$ denotes the perimeter of $D$ in $\Omega$. Under suitable technical assumptions on the data existence of solutions is proved; moreover, under local smoothness assumption on $\Gamma$ necessary optimality conditions are derived through the analysis of an adjoint problem.
[L.Ambrosio (Pavia)]

MSC 2000:: *49Q10 Optimization of the shape other than minimal surfaces
76D07 Stokes flows
35Q30 Stokes and Navier-Stokes equations
49K20 Optimal control problems with PDE (nec./ suff.)

Keywords: Navier-Stokes flow; shape optimization; necessary optimality conditions

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