A free boundary problem for the Stokes equations governing a viscous flow with over-determined condition on the free boundary is investigated. This free boundary problem is transformed into a shape optimization one which consists in minimizing a Kohn–Vogelius energy cost functional. Existence of the material derivatives of the states is proven and the corresponding variational problems are derived. Existence of the shape derivative of the cost functional is also proven and the analytic expression of the shape derivative is given in the Hadamard structure form.
Accepté le :
DOI : 10.1051/cocv/2015045
Mots-clés : Shape derivative, free boundary problems, Stokes Problem
@article{COCV_2017__23_1_195_0,
author = {Bouchon, Fran\c{c}ois and Peichl, Gunther H. and Sayeh, Mohamed and Touzani, Rachid},
title = {A free boundary problem for the {Stokes} equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {195--215},
publisher = {EDP-Sciences},
volume = {23},
number = {1},
year = {2017},
doi = {10.1051/cocv/2015045},
mrnumber = {3601021},
zbl = {1361.35209},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2015045/}
}
TY - JOUR AU - Bouchon, François AU - Peichl, Gunther H. AU - Sayeh, Mohamed AU - Touzani, Rachid TI - A free boundary problem for the Stokes equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 195 EP - 215 VL - 23 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015045/ DO - 10.1051/cocv/2015045 LA - en ID - COCV_2017__23_1_195_0 ER -
%0 Journal Article %A Bouchon, François %A Peichl, Gunther H. %A Sayeh, Mohamed %A Touzani, Rachid %T A free boundary problem for the Stokes equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 195-215 %V 23 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015045/ %R 10.1051/cocv/2015045 %G en %F COCV_2017__23_1_195_0
Bouchon, François; Peichl, Gunther H.; Sayeh, Mohamed; Touzani, Rachid. A free boundary problem for the Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 195-215. doi : 10.1051/cocv/2015045. http://archive.numdam.org/articles/10.1051/cocv/2015045/
and , Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44 (1994) 109–140. | DOI | MR | Zbl
, and , Detecting an obstacle immersed in a fluid by shape optimization methods. M3AS 21 (2011) 2069–2101. | MR | Zbl
, , , and , A Dirichlet-Neumann cost functional approach for the Bernoulli problem. J. Eng. Math. 81 (2013) 157–176. | DOI | MR | Zbl
and , Boundary conditions of a naturally permeable wall. J. Fluid Mech. 30 (1967) 197–207. | DOI
F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier–Stokes equations and related models. Springer, New York (2012). | MR | Zbl
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York (1996). | MR | Zbl
and , Structure of Shape Derivatives for Nonsmooth Domains. J. Funct. Anal. 104 (1992) 1–33. | DOI | MR | Zbl
M.C. Delfour and J.-P. Zolésio, Shapes and Geometries. SIAM (2001). | MR | Zbl
and , On a Kohn-Vogelius like formulation of free boundary problems. Comput. Optim. Appl. 52 (2012) 69–85. | DOI | MR | Zbl
G.P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations. Springer Tracts in Natural Philosophy. Springer-Verlag, New York (1994). | MR | Zbl
V. Girault and P.A. Raviart, Finite element methods for Navier–Stokes equations. Springer, Berlin (1980). | Zbl
A. Henrot and M. Pierre, Variation et Optimisation de Formes. Vol. 48 of Math. Appl. Springer (2005). | MR | Zbl
and , Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems 6 (1990) 389–414. | DOI | MR | Zbl
and , Determining conductivity by boundary measurements. Commun. Pure Appl. Math. 37 (1984) 289–298. | DOI | MR | Zbl
and , Structure of shape derivatives around irregular domains and applications. J. Convex Analysis 14 (2007) 807–822. | MR | Zbl
P. Plotnikov and J. Sokolowski, Compressible Navier–Stokes Equations. Vol. 73 of Monogr. Mat. Springer (2012). | MR | Zbl
, On the Stokes Equation with the Leak and Slip Boundary Conditions of Friction Type: Regularity of Solutions. Publ. RIMS, Kyoto Univ. 40 (2004) 345–383. | DOI | MR | Zbl
and , Study of coating flow by the finite element method. J. Comput. Phys. 42 (1981) 53–76. | DOI | Zbl
, On the free surface of a viscous fluid motion. Proc. R. Soc. London A. 349 (1976) 183–204. | DOI | MR | Zbl
and , Separating flow near a static contact line: slip at a wall and shape of a free surface. J. Comput. Phys. 34 (1980) 287–313. | DOI | MR | Zbl
J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer (1992). | MR | Zbl
, Solvability of three-dimensional problems with a free boundary for a stationary system of Navier–Stokes equations. J. Sov. Math. 21 (1983) 427–450. | DOI | Zbl
, On some free boundary problems for the Navier–Stokes equations with moving contact points and lines. Math. Ann. 302 (1995) 743–772. | DOI | MR | Zbl
V.A. Solonnikov, On the problem of a moving contact angle. Nonlinear Anal. and Cont. Mech. Papers for the 65th Birthday of James Serrin (1998) 107–137. | MR
R. Verfürth, Finite element approximation of stationary Navier–Stokes equations with slip boundary condition. Habilitationsschrift, Report No. 75, University Bochum (1986). | Zbl
, An Optimal Design Procedure for Optimal Control Support, Convex Analysis and its Application. Lect. Notes Econ. Math. Syst. 144 (1977) 207–219. | DOI | MR | Zbl
Cité par Sources :
