Geodesic
An optimal control problem for a~stationary flow of a~Jeffreys medium with slip boundary condition
Sibirskij žurnal industrialʹnoj matematiki, Tome 17 (2014) no. 1, pp. 18-27.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study an optimal control problem for the stationary motion equations of a Jeffreys viscoelastic medium with a Navier slip boundary condition. The control parameter is provided by an external force. We prove the existence of a weak solution minimizing a given cost functional and establish some properties of the solution set to the optimization problem.
Keywords: optimal control, flow control, non-Newtonian fluid, viscoelastic medium, Jeffreys model, Navier–Stokes equations, Navier slip boundary condition, weak solution, Galerkin method. 
@article{SJIM_2014_17_1_a2,
     author = {E. S. Baranovskii},
     title = {An optimal control problem for a~stationary flow of {a~Jeffreys} medium with slip boundary condition},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {18--27},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2014_17_1_a2/}
}
TY  - JOUR
AU  - E. S. Baranovskii
TI  - An optimal control problem for a~stationary flow of a~Jeffreys medium with slip boundary condition
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2014
SP  - 18
EP  - 27
VL  - 17
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2014_17_1_a2/
LA  - ru
ID  - SJIM_2014_17_1_a2
ER  - 
%0 Journal Article
%A E. S. Baranovskii
%T An optimal control problem for a~stationary flow of a~Jeffreys medium with slip boundary condition
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2014
%P 18-27
%V 17
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2014_17_1_a2/
%G ru
%F SJIM_2014_17_1_a2
E. S. Baranovskii. An optimal control problem for a~stationary flow of a~Jeffreys medium with slip boundary condition. Sibirskij žurnal industrialʹnoj matematiki, Tome 17 (2014) no. 1, pp. 18-27. http://geodesic.mathdoc.fr/item/SJIM_2014_17_1_a2/

[1] Reiner M., Reologiya, Fizmatgiz, M., 1965

[2] Astarita G., Marucci G., Principles of Non-Newtonian Fluid Hydromechanics, McGraw-Hill, N.Y., 1974

[3] Guillopé C., Saut J.-C., “Résultats d'existence pour des fluides viscoélastiques à loi de comportement de type differéntiel”, C. R. Acad. Sci. Ser. 1 Math., 305 (1987), 489–492 | MR | Zbl

[4] Renardy M., “Recent advances in the mathematical theory of steady flow of viscoelastic fluids”, J. Non-Newtonian Fluid Mech., 29 (1988), 11–24 | DOI | Zbl

[5] Renardy M., “Nonlinear stability of flows of Jeffreys fluids at low Weissenberg numbers”, Arch. Rational Mech. Anal., 132 (1995), 37–48 | DOI | MR | Zbl

[6] Turganbaev E. M., “Nachalno-kraevye zadachi dlya uravnenii vyazkouprugoizhidkosti tipa Oldroida”, Sib. mat. zhurn., 36:2 (1995), 444–458 | MR | Zbl

[7] Vorotnikov D. A., “O suschestvovanii slabykh statsionarnykh reshenii kraevoi zadachi v modeli Dzheffrisa dvizheniya vyazkouprugoi sredy”, Izv. vuzov. Matematika, 2004, no. 9, 13–17 | MR

[8] Baranovskii E. S., “Neodnorodnaya kraevaya zadacha dlya statsionarnykh uravnenii modeli Dzheffrisa dvizheniya vyazkouprugoi sredy”, Sib. zhurn. industr. matematiki, 15:3(51) (2012), 16–23

[9] Radzhagopal K. R., “O nekotorykh nereshennykh problemakh nelineinoi dinamiki zhidkostei”, Uspekhi mat. nauk, 58:2 (2003), 111–122 | DOI | MR | Zbl

[10] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR | Zbl

[11] Fursikov A. V., Optimalnoe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya, Nauch. kniga, Novosibirsk, 1999 | Zbl

[12] Navier C. L. M. H., “Memoire sur le lois du mouvement des fluides”, Mem. Acad. Roy. Sci. Paris, 6 (1823), 389–416

[13] Itoh Sh., Tanaka N., Tani A., “Steady solution and its stability for Navier–Stokes equations with general Navier slip boundary condition”, J. Math. Sci., 159:4 (2009), 472–485 | DOI | MR | Zbl

[14] Iftimie D., Sueur F., “Viscous boundary layers for the Navier–Stokes equationswith the Navier slip conditions”, Arch. Rational Mech. Anal., 199:1 (2011), 145–175 | DOI | MR | Zbl

[15] Wang L., Xin Z., Zang A., “Vanishing viscous limits for 3D Navier–Stokes equations with a Navier-slip boundary condition”, J. Math. Fluid Mech., 14:4 (2012), 791–825 | DOI | MR | Zbl

[16] Masmoudi N., Rousset F., “Uniform regularity for the Navier–Stokes equation with Navier boundary condition”, Arch. Rational Mech. Anal., 203:2 (2012), 529–575 | DOI | MR | Zbl

[17] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[18] Galdi G. P., An introduction to the mathematical theory of the Navier–Stokes equations. Steady-state problems, Springer-Verl., N.Y., 2011 | MR | Zbl

[19] Litvinov V. G., Dvizhenie nelineino-vyazkoi zhidkosti, Nauka, M., 1982 | MR | Zbl

[20] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR

[21] Lloyd N. G., Degree Theory, Univ. Press, Cambridge, 1978 | MR | Zbl

[22] Baranovskii E. S., “Oboptimalnykh zadachakh dlya sistem parabolicheskogo tipa s asferichnymi mnozhestvami dopustimykh upravlenii”, Izv. vuzov. Matematika, 2009, no. 12, 74–79 | MR | Zbl

[23] Danford N., Shvarts Dzh., Lineinye operatory. Obschaya teoriya, Izd-vo inostr. lit., M., 1962

[24] Adams R. A., Fournier J. J. F., Sobolev spaces, Elsevier, Amsterdam, 2003 | MR