Shape Optimization for Stokes Problems with Threshold Slip

Jaroslav Haslinger* and Jan Stebel

An important part of mathematical modelling of fluid flow is the choice of appropriate boundary conditions. For solid impermeable walls, traditionally no-slip boundary conditions are used. In some applications, however, one can observe a tangential velocity along the surface. To involve this effect into the mathematical model, some kind of slip boundary conditions should be utilized \cite{MR05}. Here we use a system of friction type conditions when the change from ''no-slip'' to ''slip'' status depends on the mutual relation between the shear stress and a given threshold. Due to nonsmoothness of this boundary condition, the weak formulation leads to an inequality type problem. To simplify our presentation, we restrict ourselves to the Stokes problems formulated in bounded plane domains $\Omega$. Our main goal will be to study under which conditions concerning smoothness of $\Omega$, solutions to the Stokes system with the slip boundary conditions depend continuously on variations of $\Omega$. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals \cite{HSprep}. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero. \begin{paragraph}{Acknowledgement.} The first author acknowledges the support of the grant P201/12/0671 of GAČR. The second author was supported by the grant 201/09/0917 of GAČR. \end{paragraph} \begin{thebibliography}{9} \bibitem{MR05} J.~M\'alek, K.~R.~Rajagopal: {\em Mathematical issues concerning the Navier-Stokes equations and some of its generalizations}. In Evolutionary equations. Vol. II, Handbook of Differtial Equations, Elsevier/North Holland, Amsterdam 2005, pp. 371-459. \bibitem{HSprep} J.~Haslinger, J.~Stebel: {\em Shape optimization for Stokes problems with threshold slip} (in preparation) \end{thebibliography}

Mathematics Subject Classification: 93B51

Keywords: slip boundary conditions, Stokes system, shape optimization

Minisymposion: Nonsmooth and Unilateral Problems - Modelling, Analysis and Optimization Methods