On the Global Solvability of the Two-Dimensional Through-Flow Problem for the Euler Equations with Unbounded Vorticity at the Entrance

A. E. Mamontov   ; M. I. Uvarovskaya

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 11 (2011) no. 4, pp. 69-77

Résumé

We consider the global solvability of the two-dimensional through-flow problem for an ideal incompressible fluid with restrictions for the smoothness of the input data and solution (especially, for the vorticity) as weak as possible. The normal component of the velocity is prescribed on the whole boundary of the flow domain and the vorticity is prescribed at the entrance. It is shown that the global existence theorem can be proved in the class $\{{\rm rot}\,\boldsymbol{u} \in L_{\alpha}\}$ as $\alpha>4/3$, using the regularization of the input data and compactness arguments.

Détail

Mots-clés : Euler equations, existence of solutions
Keywords: ideal incompressible fluid, nonstationary flows, nonsmooth data, through-flow problem.

Bibliographie

[1] Morgulis A. B., “O suschestvovanii dvumernykh nestatsionarnykh techenii idealnoi neszhimaemoi zhidkosti, dopuskayuschikh vikhr, ne summiruemyi so stepenyu, bolshei edinitsy”, Sib. mat. zhurn., 33:5 (1992), 209–212 | MR | Zbl

[2] Mamontov A. E., Uvarovskaya M. I., “Nestatsionarnye techeniya idealnoi neszhimaemoi zhidkosti: usloviya suschestvovaniya i edinstvennosti reshenii”, Prikl. mekh. tekhn. fiz., 49:4(490) (2008), 130–145 | MR

[3] Delort J.-M., “Existence de Nappes de Tourbillon en Dimension Deux”, J. AMS, 4:3 (1991), 553–586 | MR | Zbl

[4] Delort J.-M., “Existence de Nappes de Tourbillon sur $R^2$”, C. R. Acad. Sci. Paris, 312:1 (1991), 85–88 | MR | Zbl

[5] Delort J.-M., “Une Remarque sur le Probleme des Nappes de Tourbillon Axisymetriques sur $R^3$”, J. Func. Anal., 108 (1992), 274–295 | DOI | MR | Zbl

[6] Bardos C., “Existence et Unicité de la Solution de l'Équation d'Euler en Dimention Deux”, J. Math. Analysis and Appl., 40:3 (1972), 769–790 | DOI | MR | Zbl

[7] Lopes Filho M. C., Nussenzveig Lopes H. J., Xin Z., “Existence of Vortex Sheets with Reflection Symmetry in Two Space Dimensions”, Arch. Rat. Mech. Anal., 158:3 (2001), 235–257 | DOI | MR | Zbl

[8] Majda A., “Remarks on Weak Solutions for Vortex Sheets with a Distinguished Sign”, Ind. Univ. Math. J., 42 (1993), 921–939 | DOI | MR | Zbl

[9] Evans L. C., Müller S., “Hardy Spaces and the Two-Dimensional Euler Equations with Nonnegative Vorticity”, J. AMS, 7 (1994), 199–219 | MR | Zbl

[10] Kochin N. E., “Ob odnoi teoreme suschestvovaniya gidrodinamiki”, Prikl. mat. mekh., 20:2 (1956), 153–172

[11] Ukhovskii M. R., O razreshimosti trekhmernoi zadachi protekaniya idealnoi neszhimaemoi zhidkosti, Dep. v VINITI 27.03.79, No 1051-79, Rostov n/D, 1979

[12] Zayachkovski V., “O razreshimosti v malom odnoi nestatsionarnoi zadachi protekaniya dlya idealnoi neszhimaemoi zhidkosti”, Zap. nauch. seminarov LOMI, 96, Mat. in-t AN SSSR. Leningr. otd-nie, L., 1980, 39–56 | MR

[13] Kazhikhov A. V., “Zamechanie k postanovke zadachi protekaniya dlya uravnenii idealnoi zhidkosti”, Prikl. mat. mekh., 44:5 (1980), 947–950 | MR

[14] Kazhikhov A. V., Ragulin V. V., “Nestatsionarnye zadachi o protekanii idealnoi zhidkosti skvoz ogranichennuyu oblast”, Dokl. AN SSSR, 250:6 (1980), 1344–1347 | MR | Zbl

[15] Mamontov A. E., “On the Uniqueness of Solutions to Boundary Value Problems for Non-Stationary Euler Equations”, New Directions in Mathematical Fluid Mechanics, The Alexander V. Kazhikhov Memorial Volume, Adv. in Math. Fluid Mech., eds. A. V. Fursikov, G. P. Galdi, V. V. Pukhnachev, Birkhäuser Verlag, Basel, 2009, 281–299 | MR

[16] Antontsev S. N., Kazhikhov A. V., Monakhov V. N., Kraevye zadachi mekhaniki neodnorodnykh zhidkostei, Nauka, Novosibirsk, 1983 | Zbl

[17] Yudovich V. I., “Dvumernaya nestatsionarnaya zadacha o protekanii idealnoi neszhimaemoi zhidkosti skvoz zadannuyu oblast”, Mat. sb., 64 (106):4 (1964), 562–588

[18] Ukhovskii M. R., “Ob osesimmetricheskoi zadache s nachalnymi dannymi dlya uravnenii dvizheniya idealnoi neszhimaemoi zhidkosti”, Mekhanika zhidkosti i gaza, 1967, no. 3, 3–12

[19] Alekseev G. V., “O razreshimosti neodnorodnoi kraevoi zadachi dlya dvumernykh nestatsionarnykh uravnenii dinamiki idealnoi zhidkosti”, Dinamika zhidkosti so svobodnymi granitsami, Dinamika sploshnoi sredy, 24, Novosibirsk, 1976, 15–35

[20] Alekseev G. V., “Ob ischezayuschei vyazkosti v dvumernykh statsionarnykh zadachakh gidrodinamiki neszhimaemoi zhidkosti”, Sb. nauch. tr., Dinamika sploshnoi sredy, 10, AN SSSR. Sib. otd-nie. In-t gidrodinamiki, Novosibirsk, 1972, 5–27 | MR

[21] Alekseev G. V., “O edinstvennosti i gladkosti ploskikh vikhrevykh techenii idealnoi zhidkosti”, Sb. nauch. tr., Dinamika sploshnoi sredy, 15, AN SSSR. Sib. otd-nie. In-t gidrodinamiki, Novosibirsk, 1973, 7–17 | MR

[22] Morgulis A. B., “Razreshimost trekhmernoi statsionarnoi zadachi protekaniya”, Sib. mat. zhurn., 40:1 (1999), 142–158 | MR | Zbl

[23] Alekseev G. V., “O stabilizatsii reshenii dvumernykh uravnenii dinamiki idealnoi zhidkosti”, Prikl. mekh. tekhn. fiz., 1977, no. 2(102), 85–92

[24] Kazhikhov A. V., Izbrannye trudy. Matematicheskaya gidrodinamika, Izd-vo In-ta gidrodinamiki im. M. A. Lavrenteva SO RAN, Novosibirsk, 2008

[25] Kazhikhov A. V., Ragulin V. V., “O zadache protekaniya dlya uravnenii idealnoi zhidkosti”, Zap. nauch. seminarov LOMI, 96, Mat. in-t AN SSSR. Leningr. otd-nie, L., 1980, 84–96 | MR | Zbl

[26] Alekseev G. V., “O suschestvovanii edinstvennogo techeniya provodyaschei zhidkosti v slabo iskrivlennom kanale”, Sb. nauch. tr., Dinamika sploshnoi sredy, 3, AN SSSR. Sib. otd-nie. In-t gidrodinamiki, Novosibirsk, 1969, 115–121

[27] Kazhikhov A. V., “Korrektnost nestatsionarnoi zadachi o protekanii idealnoi zhidkosti cherez zadannuyu oblast”, Sb. nauch. tr., Dinamika sploshnoi sredy, 47, AN SSSR. Sib. otd-nie. In-t gidrodinamiki, Novosibirsk, 1980, 37–56 | MR

[28] Kazhikhov A. V., “Dvumernaya zadacha o protekanii idealnoi zhidkosti cherez zadannuyu oblast”, Kraevye zadachi dlya neklassicheskikh UMF, Novosibirsk, 1989, 32–37 | MR

[29] Kazhikhov A. V., “Nachalno-kraevye zadachi dlya uravnenii Eilera neszhimaemoi zhidkosti”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 1991, no. 5, 13–19 | MR

[30] Tsygankova E. K., “Ob odnoi novoi postanovke dvumernoi zadachi protekaniya zhidkosti cherez zadannuyu oblast”, Vestn. Novosib. gos. un-ta. Seriya: Matematika, mekhanika, informatika, 4:3/4 (2004), 93–99 | Zbl

[31] Kazhikhov A. V., “Ob odnom podkhode k kraevym zadacham dlya uravnenii sostavnogo tipa”, Sib. mat. zhurn., 33:6 (1992), 47–53 | MR

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

[33] Aubin J. P., “Une Théorème de Compacité”, C. R. Acad. Sc., 256 (1963), 5042–5044 | MR | Zbl

[34] Simon J., “Compact Sets in the Space $L_p(0,T;B)$”, Ann. Mat. Pura Appl., 146 (1987), 65–96 | DOI | MR | Zbl