Previous |  Up |  Next


Stokes problem; friction boundary condition; shape optimization
We study the Stokes problems in a bounded planar domain $\Omega $ with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning the 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. 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.
[1] Březina, J.: Asymptotic properties of solutions to the equations of incompressible fluid mechanics. J. Math. Fluid Mech. 12 536-553 (2010). DOI 10.1007/s00021-009-0301-x | MR 2749442 | Zbl 1270.35336
[2] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics 15 Springer, New York (1991). MR 1115205 | Zbl 0788.73002
[3] Bucur, D., Feireisl, E., Nečasová, Š.: Influence of wall roughness on the slip behaviour of viscous fluids. Proc. R. Soc. Edinb., Sect. A, Math. 138 957-973 (2008). DOI 10.1017/S0308210507000376 | MR 2477446 | Zbl 1151.76004
[4] Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44 2756-2801 (2012). DOI 10.1137/110830289 | MR 3023393 | Zbl 1256.35074
[5] Chenais, D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 189-219 (1975). DOI 10.1016/0022-247X(75)90091-8 | MR 0385666 | Zbl 0317.49005
[6] Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku 888 199-216 (1994). MR 1338892 | Zbl 0939.76527
[7] Fujita, H.: A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149 57-69 (2002). DOI 10.1016/S0377-0427(02)00520-4 | MR 1952966 | Zbl 1058.76023
[8] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. 1: Linearized Steady Problems. Springer Tracts in Natural Philosophy 38 Springer, New York (1994). MR 1284206
[9] Haslinger, J., Mäkinen, R. A. E.: Introduction to Shape Optimization: Theory, Approximation, and Computation. Advances in Design and Control 7. SIAM Society for Industrial and Applied Mathematics Philadelphia (2003). MR 1969772 | Zbl 1020.74001
[10] Haslinger, J., Outrata, J. V., Pathó, R.: Shape optimization in 2{D} contact problems with given friction and a solution-dependent coefficient of friction. Set-Valued Var. Anal. 20 31-59 (2012). MR 2886504 | Zbl 1242.49088
[11] Hlaváček, I., Mäkinen, R.: On the numerical solution of axisymmetric domain optimization problems. Appl. Math., Praha 36 284-304 (1991). MR 1113952 | Zbl 0745.65044
[12] Liakos, A.: Weak imposition of boundary conditions in the Stokes and Navier-Stokes equation. PhD thesis, University of Pittsburgh (1999). MR 2700238
[13] Málek, J., Rajagopal, K. R.: Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. C. M. Dafermos Evolutionary Equations. Vol. II Handbook of Differential Equations Elsevier/North-Holland, Amsterdam (2005), 371-459. MR 2182831 | Zbl 1095.35027
[14] Navier, C. L.: Mémoire sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Paris 6 389-416 (1823).
[15] Saito, N.: On the Stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions. Publ. Res. Inst. Math. Sci. 40 345-383 (2004), errata ibid. 48 475-476 (2012). DOI 10.2977/prims/1145475807 | MR 2049639 | Zbl 1244.35061
[16] Sokołowski, J., Zolesio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics 16 Springer, Berlin (1992). DOI 10.1007/978-3-642-58106-9_1 | MR 1215733 | Zbl 0761.73003
[17] Stebel, J.: On shape stability of incompressible fluids subject to Navier's slip condition. J. Math. Fluid Mech. 14 575-589 (2012). DOI 10.1007/s00021-011-0086-6 | MR 2964751 | Zbl 1254.76058
Partner of
EuDML logo