Far Eastern Mathematical Journal

To content of the issue


The solvability of stationary boundary value problem for Euler equations


Illarionov A.A., Illarionova L.V.

2018, issue 1, Ń. 48-55


Abstract
One consider boundary value problem for 2D Euler equations describing stationary flow of ideal incompressible nonhomogeneous fluid. The boundary value of normal component of the flow velocity, and value of total pressure, density on the inflow (outflow) part of the boundary are given. We prove global solvability of the problem.

Keywords:
extraneous mobile object, Autonomous underwater vehicle, the probability of detection, the problem of Buffon

Download the article (PDF-file)

References

[1] N.E. Kochin, “Ob odnoj teoreme suwestvovanija gidrodinamiki”, Prikl. mat. meh., 20:2, (1956), 153–172.
[2] V.I. JUdovich, “Nestacionarnye techenija ideal'noj neszhimaemoj zhidkosti”, ZHurn. vych. mat. i mat. fiz., 3:6, (1963), 1032–1066.
[3] V.I. JUdovich, “Dvumernaja nestacionarnaja zadacha o protekanii ideal'noj neszhimaemoj zhidkosti skvoz' zadannuju oblast'”, Matem. sb., 106:4, (1964), 562–588.
[4] M.R. Uhovskij, “Ob osesimmetrichnoj zadache s nachal'nymi dannymi dlja uravnenij dvizhenija ideal'noj neszhimaemoj zhidkosti”, Mehanika zhidkosti i gaza, 1967, ą 3, 3–12.
[5] G.V. Alekseev, Matematicheskie voprosy teorii dvumernyh nepotencial'nyh techenij neszhimaemoj zhidkosti, diss. ... kand. fiz.-matem. nauk, Novosibirsk, 1972, 131 s.
[6] G.V. Alekseev, “O razreshimosti neodnorodnoj kraevoj zadachi dlja dvumernyh nestacionarnyh uravnenij dinamiki ideal'noj zhidkosti”, Dinamika sploshnoj sredy, 24, Novosibirsk, 1976, 15–35.
[7] A.V. Kazhihov, “Zamechanie k postanovke zadachi protekanija dlja uravnenij ideal'noj zhidkosti”, Prikl. mat. meh., 44:5, (1980), 947–950.
[8] A.V. Kazhihov, V.V. Ragulin, “Nestacionarnye zadachi o protekanii ideal'noj zhidkosti skvoz' ogranichennuju oblast'”, Dokl. AN SSSR, 250:6, (1980), 1344–1347.
[9] A.V. Kazhihov, V.V. Ragulin, “O zadache protekanija dlja uravnenij ideal'noj zhidkosti”, Zap. nauch. seminarov LOMI, 96, (1980), 84–96.
[10] A.V. Kazhihov, “Korrektnost' nestacionarnoj zadachi o protekanii ideal'noj zhidkosti cherez zadannuju oblast'”, Dinamika sploshnoj sredy, 47, Novosibirsk, 1980, 37–56.
[11] S.N. Antoncev, A.V. Kazhihov, V.N. Monahov, Kraevye zadachi mehaniki neodnorodnyh zhidkostej, Nauka, Novosibirsk, 1983.
[12] A.V. Kazhihov, “Dvumernaja zadacha o protekanii ideal'noj zhidkosti cherez zadannuju oblast'”, Kraevye zadachi dlja neklassicheskih UMF, Novosibirsk, 1989, 32–37.
[13] A.V. Kazhihov, “Nachal'no-kraevye zadachi dlja uravnenij JEjlera neszhimaemoj zhidkosti”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mehanika, 1991, ą 5, 13–19.
[14] A.E. Mamontov, M.I. Uvarovskaja, “Nestacionarnye techenija ideal'noj neszhimaemoj zhidkosti: uslovija suwestvovanija i edinstvennosti reshenij”, Prikl. meh. tehn. fiz., 49:490, (2008), 130–145.
[15] A.E. Mamontov, “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., red. A.V. Fursikov, G.P. Galdi, V.V. Pukhnachev, Birkhauser Verlag, Basel, 2009, 281–299.
[16] M.I. Uvarovskaja, Primenenie prostranstv Orlicha v zadachah dinamiki ideal'noj neszhimaemoj zhidkosti, diss. ... kand. fiz.-matem. nauk, Novosibirsk, 2009, 63 s.
[17] G.V. Alekseev, “O suwestvovanii edinstvennogo techenija provodjawej zhidkosti v slabo iskrivlennom kanale”, Dinamika sploshnoj sredy, 3, AN SSSR. Sib. otd-nie. In-t gidrodinamiki, Novosibirsk, 1969, 115–121.
[18] G.V. Alekseev, “Ob ischezajuwej vjazkosti v dvumernyh stacionarnyh zadachah gidrodinamiki neszhimaemoj zhidkosti”, Dinamika sploshnoj sredy, 10, AN SSSR. Sib. otd-nie. In-t gidrodinamiki, Novosibirsk, 1972, 5–27.
[19] G.V. Alekseev, “O edinstvennosti i gladkosti ploskih vihrevyh techenij ideal'noj zhidkosti”, Dinamika sploshnoj sredy, 15, AN SSSR. Sib. otd-nie. In-t gidrodinamiki, Novosibirsk, 1973, 7–17.
[20] O.V. Troshkin, “O topologicheskom analize struktury gidrodinamicheskih techenij”, UMN, 43:4, (1988), 129–158.
[21] O.V. Troshkin, “Dvumernaja zadacha o protekanii dlja stacionarnyh uravnenij JEjlera”, Matem. sb., 180:3, (1989), 354–374.
[22] A.B. Morgulis, “Razreshimost' trehmernoj stacionarnoj zadachi protekanija”, Sib. mat. zhurn., 40:1, (1999), 142–158.
[23] N.N. Frolov, “O razreshimosti kraevoj zadachi dvizhenija neodnorodnoj zhidkosti”, Matem. zametki, 53:6, (1993), 650–656.
[24] N.N. Frolov, “Kraevaja zadacha, opisyvajuwaja dvizhenie neodnorodnoj zhidkosti”, Sib. matem. zhurn., 37:2, (1996), 433–451.
[25] A.JU. CHebotarev, “Stacionarnye variacionnye neravenstva v modeli neodnorodnoj neszhimaemoj zhidkosti”, Sib. matem. zhurn., 38:5, (1997), 1184–1193.
[26] A.A. Illarionov, “Optimal'noe granichnoe upravlenie stacionarnym techeniem vjazkoj neodnorodnoj neszhimaemoj zhidkosti”, Matem. zametki, 69:5, (2001), 666–678.
[27] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Publ., Boston, 1985.
[28] D. Gilbarg, M. Trudinger, JEllipticheskie differencial'nye uravnenija s chastnymi proizvodnymi vtorogo porjadka, Nauka, M., 1989, 464 s.

To content of the issue