Nonlocal solutions of the Cauchy problem in scales of analytic polyalgebras

S. S. Titov

Trudy Instituta matematiki i mehaniki, Fluid dynamics, Tome 9 (2003) no. 2, pp. 105-128

Bibliographie

[1] Sidorov A. F., Izbrannye trudy. Matematika. Mekhanika, Fizmatlit, M., 2001, 576 pp. | MR

[2] Titov S. S., Reshenie uravnenii s osobennostyami v analiticheskikh shkalakh banakhovykh prostranstv, Preprint, UralGAKhA, Ekaterinburg, 1999, 264 pp. | Zbl

[3] Titov S. S., “Prostranstvenno-periodicheskie resheniya polnoi sistemy Nave–Stoksa”, Dokl. RAN, 365:6 (1999), 761–763 | Zbl

[4] Titov S. S., “Reshenie nelineinykh uravnenii v analiticheskikh polialgebrakh. I”, Izv. vuzov. Matematika, 2000, no. 1, 66–76 | MR | Zbl

[5] Titov S. S., “Reshenie nelineinykh uravnenii v analiticheskikh polialgebrakh. II”, Izv. vuzov. Matematika, 2000, no. 6, 45–52 | MR | Zbl

[6] Ovsyannikov L. V., Analiticheskie gruppy, NGU, Novosibirsk, 1972, 237 pp. | MR

[7] Ovsyannikov L. V., “Singulyarnyi operator v banakhovykh shkalakh”, Dokl. AN SSSR, 163:4 (1965), 819–822 | MR | Zbl

[8] Ovsyannikov L. V., “Nelineinaya zadacha Koshi v shkale banakhovykh prostranstv”, Dokl. AN SSSR, 200:4 (1971), 789–792 | Zbl

[9] Ovsyannikov L. V., Makarenko N. I., Nalimov V. I. i dr., Nelineinye problemy teorii poverkhnostnykh i vnutrennikh voln, Nauka, Novosibirsk, 1985, 318 pp.

[10] Kalyakin L. A., “Dlinnovolnovye asimptotiki reshenii nelineinykh uravnenii s dispersiei”, Dokl. AN SSSR, 228:4 (1986), 809–813 | MR

[11] Kowalewski S., “Zur Theorie der partiellen Differentialgleichungen”, J. Reine und Angew. Math., 80:1 (1875), 1–32

[12] Kovalevskaya S. V., Nauchnye trudy. K teorii differentsialnykh uravnenii v chastnykh proizvodnykh, Izd. AN SSSR, M., 1948, 368 pp.

[13] Titov S. S., “Zadacha Koshi v spetsialnykh shkalakh banakhovykh prostranstv”, Priblizhennye metody issled. nelinein. zadach mekhaniki splosh. sredy, IMM UrO RAN, Sverdlovsk, 1992, 51–57

[14] Titov S. S., “Analog teoremy Kovalevskoi dlya lineinykh evolyutsionnykh uravnenii”, Sib. mat. zhurn., 40:6 (1999), 1377–1379 | MR | Zbl

[15] Salekhov G. S., Fridlender V. R., “K voprosu o zadache, obratnoi k zadache Koshi–Kovalevskoi”, Uspekhi mat. nauk, 7:5(51) (1952), 169–192 | MR

[16] Le Roux L., “Sur les intégrates analytiques de l'équation $\frac{\partial^2z}{\partial y^2}=\frac{\partial z}{\partial x}$”, Bull. Sci. Math., 19 (1895), 127–129

[17] Riquier Ch., Les systèmes d'équation aux dérivées partielles, Gauthier-Villars, Paris, 1910, 590 pp.

[18] Bautin S. P., Zadacha Rike i kharakteristicheskaya zadacha Koshi, UrGUPS, Ekaterinburg, 1999, 76 pp.

[19] Titov S. S., “Analitichnost lineinykh odnoparametricheskikh grupp Li–Beklunda”, Differentsialnye uravneniya, 36:4 (1990), 699–702 | MR

[20] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983, 280 pp. | MR

[21] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978, 339 pp. | MR

[22] Titov S. S., “O zadache Koshi v banakhovykh shkalakh s kompaktnymi vlozheniyami”, Sib. mat. zhurn., 37:5 (1996), 1167–1175 | MR | Zbl

[23] Ovsyannikov L. V., “O kanonicheskoi strukture beskonechnykh grupp Li”, Tr. mezhdunar. simpoz. “Teoretiko-gruppovye metody v mekhanike”, Novosibirsk, 1978, 188–202

[24] Kaptsov O. V., “Obraschenie kanonicheskogo umnozheniya v $A$-gruppakh”, Dinamika splosh. sredy, 50, IG SO AN SSSR, Novosibirsk, 1981, 74–84 | MR

[25] Nirenberg L., “An abstract form of the nonlinear Cauchy–Kowalewski theorem”, J. Different. Geom., 6 (1972), 561–576 | MR | Zbl

[26] Nishida T., “A note on a theorem of Nirenberg”, J. Different. Geom., 12 (1977), 629–633 | MR | Zbl

[27] Titov S. S., “Veschestvennyi analog teoremy L. V. Ovsyannikova dlya dispersionnykh operatorov”, Tr. mezhdunar. konf. “Kompleks. analiz, differents. uravneniya i smezh. voprosy” II. Differents. uravneniya, Izd-vo IM s VTs UNTs RAN, Ufa, 2000, 140–145

[28] Bryuno A. D., Stepennaya geometriya v algebraicheskikh i differentsialnykh uravneniyakh, Nauka, M., 1998, 288 pp. | MR | Zbl

[29] Melnikov B. N., Melnikov Yu. B., Problemy metodologii issledovaniya geotekhnogennykh struktur, IGG UrO RAN, Ekaterinburg, 1998, 303 pp.

[30] Pilimonov M. Yu., Korzunin L. G., Sidorov A. F., “Approximate methods for solving nonlinear initial boundary-value problems based on special construction of series”, Rus. J. Numer. Anal. and Math. Modelling, 8:2 (1993), 101–125 | MR

[31] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989, 336 pp. | MR

[32] Titov S. S., “Reshenie periodicheskikh zadach Koshi s pomoschyu spetsialnykh trigonometricheskikh ryadov”, Chisl. metody mekhaniki splosh. sredy, 9:2 (1978), 112–124, Novosibirsk | MR

[33] Grauert G., Remmert R., Analiticheskie lokalnye algebry, Nauka, M., 1988, 304 pp. | MR | Zbl

[34] Titov S. S., “Veschestvennye analogi teoremy S. V. Kovalevskoi v mekhanike vyazkoi zhidkosti”, Simposium of PDE and special topics of theory of ODE dedic. 150-th arm. of birthday of S. V. Kovalevskaya, St. Petersburg, 2000, 31–33

[35] Titov S. S., “Suschestvovanie periodicheskikh techenii vyazkoi neszhimaemoi zhidkosti v tselom po nachalnym dannym”, Tr. 31-i konf. “Problemy teoret. i prikl. matematiki”, Ekaterinburg, 2000, 69–73

[36] Titov S. S., “Asimmetrichnye resheniya sistemy Byurgersa s usloviem prilipaniya”, Tr. mezhdunar. konf. “Simmetriya i differentsialnye uravneniya”, Krasnoyarsk, 2000, 221–223 | Zbl

[37] Sidorov A. F., Khairullina O. B., “Raschet geksagonalnoi konvektsii v yacheikakh Benara s pomoschyu spetsialnykh trigonometricheskikh ryadov”, Priblizhennye metody issledovaniya nelinein. zadach mekhaniki splosh. sredy, Sverdlovsk, 1985, 35–50

[38] Marchuk G. I., Agoshkov V. I., Vvedenie v proektsionno-setochnye metody, Nauka, M., 1981, 416 pp. | MR

[39] Sidorov A. F., Shapeev V. P., Yanenko N. N., Metod differentsialnykh svyazei i ego primenenie v gazovoi dinamike, Nauka, Novosibirsk, 1984, 272 pp. | MR

[40] Nishida T., Kawashima S., Matsumura A., “On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier–Stokes equation”, Commun. Math. Phys., 70:2 (1979), 97–124 | DOI | MR | Zbl

[41] Nishida T., Matsumura A., “The initial value problem for the equations of motion of viscous and heat conductive gases”, J. Math. Kyoto Univ., 20:1 (1980), 67–104 | MR | Zbl

[42] Galkin V. A., “Funktsionalnye resheniya zakonov sokhraneniya”, Dokl. AN SSSR, 310:4 (1990), 834–839 | MR | Zbl

[43] Treves F., “An abstract nonlinear Cauchy–Kovalevska theorem”, Trans. Amer. Math. Soc., 150 (1970), 77–92 | DOI | MR | Zbl

[44] Duchateau P., Treves J. P., “An abstract Cauchy–Kowalevska theorem in scales of Gevrey classes”, Symposia Math., Vol. 7, Acad. Press, N.-Y., 1971, 135–163 | MR

[45] Temam R., The Navier–Stokes Equations, The Theory and Numerical Analysis, North-Holland publ. Co., N.-Y., 1979, 519 pp. | MR | Zbl

[46] Antontsev C. H., Kazhikhov A. V., Monakhov B. H., Kraevye zadachi mekhaniki neodnorodnykh zhidkostei, Nauka, N., 1983, 319 pp. | Zbl

[47] Lipman G. V., Roshko A., Elementy gazovoi dinamiki, Izd-vo inostr. lit., M., 1960, 518 pp. | MR

[48] Nash J., “Le problème de Cauchy pour les équations differentielles d'un fluide génèral”, Bull. Soc. Math. France, 90:4 (1962), 486–497 | MR

[49] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Nauka, M., 1970, 288 pp. | MR

[50] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1976, 736 pp. | MR

[51] Solonnikov V. A., “O razreshimosti nachalno-kraevoi zadachi dlya uravneniya dvizheniya vyazkoi szhimaemoi zhidkosti”, Issled. po linein. operatoram i teorii funktsii. VI, Zap. nauch. seminarov LOMI, 56, Nauka, Leningrad, 1976, 128–142 | MR | Zbl

[52] Solonnikov V. A., “O zadachakh gidrodinamiki vyazkoi neszhimaemoi zhidkosti v oblastyakh s nekompaktnymi granitsami”, Algebra i analiz, 4:6 (1992), 28–53 | MR | Zbl

[53] Vaigant V. A., Kazhikhov A. V., “O suschestvovanii globalnykh reshenii dvumernykh uravnenii Nave–Stoksa szhimaemoi vyazkoi zhidkosti”, Sib. mat. zhurn., 36:6 (1995), 1283–1316 | MR

[54] Mamontov A. E., “O globalnoi razreshimosti mnogomernykh uravnenii Nave–Stoksa szhimaemoi nelineino vyazkoi zhidkosti”, Sib. mat. zhurn., 40:2 (1999), 408–420 | MR | Zbl

[55] Bautin S. P., “Ob odnom klasse reshenii sistemy Nave–Stoksa, periodicheskikh po prostranstvennym peremennym”, Modelirovanie v mekhanike, 1:1 (1987), 3–15, Novosibirsk, VTs i ITPM SO AN SSSR | MR | Zbl

[56] Bautin S. P., “Analiticheskoe postroenie techenii vyazkogo gaza s pomoschyu posledovatelnosti linearizovannykh sistem Nave–Stoksa”, Prikl. matematika i mekhanika, 52:4 (1988), 579–589 | MR | Zbl

[57] Shatov A. K., “O zadache Koshi dlya linearizovannoi sistemy uravnenii Nave–Stoksa s uchetom szhimaemosti”, Dokl. AN SSSR, 285:6 (1985), 1374–1376 | MR | Zbl

[58] Titov S. S., “Metod assotsiativnykh kolets dlya resheniya uravnenii matematicheskoi fiziki”, Tochnye i priblizhennye metody resheniya zadach mekhaniki splosh. sredy, UNTs AN SSSR, Sverdlovsk, 1982, 93–96