Geodesic
On solutions of the Gol'dshtik problem
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 4, pp. 409-415.

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

The Gol'dshtik model for separated flows of incompressible fluid is considered. A solution of the given two-dimensional problem in mathematical physics for a finite domain is found with the finite element method. Estimations of the differential operator are obtained. A result on the number of solutions of the Gol'dshtik problem is obtained using the variational method. 
@article{SJVM_2012_15_4_a5,
     author = {D. K. Potapov},
     title = {On solutions of the {Gol'dshtik} problem},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {409--415},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2012_15_4_a5/}
}
TY  - JOUR
AU  - D. K. Potapov
TI  - On solutions of the Gol'dshtik problem
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2012
SP  - 409
EP  - 415
VL  - 15
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2012_15_4_a5/
LA  - ru
ID  - SJVM_2012_15_4_a5
ER  - 
%0 Journal Article
%A D. K. Potapov
%T On solutions of the Gol'dshtik problem
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2012
%P 409-415
%V 15
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2012_15_4_a5/
%G ru
%F SJVM_2012_15_4_a5
D. K. Potapov. On solutions of the Gol'dshtik problem. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 4, pp. 409-415. http://geodesic.mathdoc.fr/item/SJVM_2012_15_4_a5/

[1] Goldshtik M. A., “Matematicheskaya model otryvnykh techenii neszhimaemoi zhidkosti”, Dokl. AN SSSR, 147:6 (1962), 1310–1313

[2] Kuiper H. J., “On positive solutions of nonlinear elliptic eigenvalue problems”, Rend. Circ. Mat. Palermo Ser. 2, 20:2–3 (1971), 113–138 | DOI | MR

[3] Fraenkel L. E., Berger M. S., “A global theory of steady vortex rings in an ideal fluid”, Acta Math., 132:1 (1974), 13–51 | DOI | MR | Zbl

[4] Chang K. C., “Free boundary problems and the set-valued mappings”, J. Differential Eq., 49:1 (1983), 1–28 | DOI | MR | Zbl

[5] Lavrentev M. A., Shabat B. V., Problemy gidrodinamiki i ikh matematicheskie modeli, Nauka, M., 1973 | MR

[6] Vainshtein I. I., Yurovskii V. K., “Ob odnoi zadache sopryazheniya vikhrevykh techenii idealnoi zhidkosti”, Zhurn. prikl. mekh. i tekhn. fiziki, 1976, no. 5, 98–100

[7] Titov O. V., “Variatsionnyi podkhod k ploskim zadacham o skleike potentsialnogo i vikhrevogo techeniya”, Prikl. matem. i mekhanika, 41:2 (1977), 370–372

[8] Potapov D. K., “Matematicheskaya model otryvnykh techenii neszhimaemoi zhidkosti”, Izvestiya RAEN. Ser. MMMIU, 8:3–4 (2004), 163–170

[9] Vainshtein I. I., Litvinov P. S., “Model M. A. Lavrenteva o skleike vikhrevykh i potentsialnykh techenii idealnoi zhidkosti”, Vestn. SibGAU, 2009, no. 3(24), 7–9

[10] Potapov D. K., “Nepreryvnye approksimatsii zadachi Goldshtika”, Matem. zametki, 87:2 (2010), 262–266 | DOI | MR | Zbl

[11] Vainshtein I. I., “Dualnaya zadacha k zadache M. A. Goldshtika s proizvolnoi zavikhrennostyu”, Zhurn. SFU. Ser. matem. i fizika, 3:4 (2010), 500–506

[12] Potapov D. K., “Upravlenie spektralnymi zadachami dlya uravnenii s razryvnymi operatorami”, Tr. IMM UrO RAN, 17, no. 1, 2011, 190–200

[13] Potapov D. K., “Bifurkatsionnye zadachi dlya uravnenii ellipticheskogo tipa s razryvnymi nelineinostyami”, Matem. zametki, 90:2 (2011), 280–284 | DOI | MR | Zbl

[14] Potapov D. K., “Nepreryvnaya approksimatsiya odnomernogo analoga modeli Goldshtika otryvnykh techenii neszhimaemoi zhidkosti”, Sib. zhurn. vychisl. matematiki (RAN. Sib. otd-nie, Novosibirsk), 14:3 (2011), 291–296

[15] Vainshtein I. I., “Reshenie dvukh dualnykh zadach o skleike vikhrevykh i potentsialnykh techenii variatsionnym metodom M. A. Goldshtika”, Zhurn. SFU. Ser. matem. i fizika, 4:3 (2011), 320–331

[16] Potapov D. K., “Ob odnom klasse ellipticheskikh variatsionnykh neravenstv so spektralnym parametrom i razryvnoi nelineinostyu”, Sib. mat. zhurnal, 53:1 (2012), 205–212 | MR | Zbl

[17] Potapov D. K., “Zadachi upravleniya dlya uravnenii so spektralnym parametrom i razryvnym operatorom pri nalichii vozmuschenii”, Zhurn. SFU. Ser. matem. i fizika, 5:2 (2012), 239–245

[18] Pavlenko V. N., Potapov D. K., “O suschestvovanii lucha sobstvennykh znachenii dlya uravnenii s razryvnymi operatorami”, Sib. mat. zhurnal, 42:4 (2001), 911–919 | MR | Zbl

[19] Krasnoselskii M. A., Pokrovskii A. V., “Pravilnye resheniya uravnenii s razryvnymi nelineinostyami”, Dokl. AN SSSR, 226:3 (1976), 506–509 | MR

[20] Potapov D. K., “Ob odnoi otsenke sverkhu velichiny bifurkatsionnogo parametra v zadachakh na sobstvennye znacheniya dlya uravnenii ellipticheskogo tipa s razryvnymi nelineinostyami”, Diff. uravneniya, 44:5 (2008), 715–716 | MR | Zbl

[21] Potapov D. K., “Otsenki differentsialnogo operatora v zadachakh so spektralnym parametrom dlya uravnenii ellipticheskogo tipa s razryvnymi nelineinostyami”, Vestn. Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Ser. fiz.-mat. nauki, 2010, no. 5(21), 268–271

[22] Chang K. C., “Variational methods for non-differentiable functionals and their applications to partial differential equations”, J. Math. Anal. and Appl., 80:1 (1981), 102–129 | DOI | MR | Zbl