Development of Studies of the School of L.I. Sedov at the Chair of Hydromechanics

A. V. Aksenov   ; A. N. Golubyatnikov

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2024), pp. 88-93

Résumé

An overview of recent works in the field of gas dynamics and plasma dynamics, which were initiated by academician L.I. Sedov and his followers, is given. An analytical study of the equations was carried out, exact solutions were constructed and the problem of energy–momentum concentration was solved. Higher invariants of characteristics for a system of equations of one-dimensional gas dynamics in Eulerian and Lagrangian variables for special adiabatic exponents are found. Based on the use of higher invariants of characteristics, the solution of the Cauchy problem is reduced to a system of ordinary differential equations. Two Cauchy problems are presented, the solutions of which exist indefinitely without a gradient catastrophe.

Bibliographie

[1] Sedov L.I., Metody podobiya i razmernosti v mekhanike, Nauka, M., 1981 | MR

[2] Sedov L.I., “Ob odnom vozmozhnom mekhanizme porozhdeniya bezmassovykh chastits”, Dokl. AN SSSR, 285:1, 350–353 | MR

[3] Shapiro P.L., “Relativistic blast waves in two dimensions. I. The adiabatic case”, J. Astrophys., 233:3 (1979), 831–850 | DOI

[4] Golubyatnikov A.N., “O mekhanizme otdeleniya energii-impulsa ot massy pokoya”, Mekhanika. Sovremennye problemy, Sb. statei, Izd-vo MGU, M., 1987, 152–157

[5] Golubyatnikov A.N., Plotnikov S.I., “Ob otdelenii energii ot massy v neodnorodnykh diskretnykh mekhanicheskikh sistemakh”, Dokl. AN SSSR, 309:3 (1989), 559–562 | Zbl

[6] Golubyatnikov A.N., “Ob uskorenii udarnykh voln i kontsentratsii energii”, Tr. Matem. in-ta RAN, 281, 2013, 262–269

[7] Bogoyavlenskii O.I., Metody kachestvennoi teorii dinamicheskikh sistem v astrofizike i gazovoi dinamike, Nauka, M., 1980 | MR

[8] Golubyatnikov A.N., Ukrainskii D.V., “Odno tochnoe reshenie ob obzhatii polosti v vyazkoi teploprovodnoi szhimaemoi srede”, Izv. RAN. Mekhan. zhidkosti i gaza, 2022, no. 4, 77–85 | Zbl

[9] Golubyatnikov A.N., Ukrainskii D.V., “K probleme kontsentratsii energii”, Tr. Matem. in-ta RAN, 322, 2023, 83–93 | DOI | Zbl

[10] Zeldovich Ya.B., Novikov I.D., Relyativistskaya astrofizika, Nauka, M., 1967

[11] Alfven H., Klein O., “Matter–antimatter annihilation and cosmology”, Arkiv für fysik, 23:19 (1962), 187–194

[12] Bonnevier B., “On the early development of the Metagalactic System”, Arkiv für fysik, 27:21 (1964), 305–310

[13] Alven Kh., “Antiveschestvo i evolyutsiya Metagalaktiki”, Astron. zhurn., 42:5 (1965), 873–892

[14] Golubiatnikov A.N., Lyuboshits D.B., “On strong shock waves of annihilation”, Gravitation and Cosmology, 22:3 (2016), 258–263 | DOI | MR | Zbl

[15] Golubiatnikov A.N., Lyuboshits D.B., “A cosmological solution with acceleration caused by an annihilation shock wave”, Gravitation and Cosmology, 25:2 (2019), 131–137 | DOI | MR | Zbl

[16] Golubyatnikov A.N., Zakharov S.A., “O modelirovanii protsessa annigilyatsii v sobstvennom gravitatsionnom pole”, Mezhdunar. nauch. konf. “Matematika v sozvezdii nauk”. K yubileyu rektora MGU akad. V.A. Sadovnichego, Tez. dokl. (Moskva, 1–2 aprelya 2024), M., 2024, 438–440

[17] Golubyatnikov A.N., “Razryvy maloi amplitudy reshenii uravnenii mekhaniki sploshnoi sredy”, Tr. Matem. in-ta RAN, 300, 2018, 65–75 | DOI | Zbl

[18] Riemann B., “Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite”, Aus dem achten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematischphysikalische Klasse, 8:43 (1860), 43–66 https://eudml.org/doc/135717

[19] Darboux J.G., “Sur la théorie des équations aux dérivées partielles du second ordre”, C. r. Acad. sci. A, 70 (1870), 675–678 | MR

[20] Darboux J.G., “Sur la théorie des équations aux dérivées partielles”, C. r. Acad. sci. A, 70 (1870), 746–749

[21] Goursat M.E., Leçons sur l'intégration des équations aux deréeves partielles du second order a deux variables independantes, v. II, Paris, 1898

[22] Kaptsov O.V., Metody integrirovaniya uravnenii s chastnymi proizvodnymi, Fizmatlit, M., 2009

[23] Aksenov A.V., “Nelineinye periodicheskie volny v gaze”, Izv. RAN. Mekhan. zhidkosti i gaza, 2012, no. 5, 88–98 | Zbl

[24] Aksenov A.V., “Simmetrii i sootnosheniya mezhdu resheniyami klassa uravnenii Eilera–Puassona–Darbu”, Dokl. RAN, 381:2 (2001), 176–179 | MR

[25] Aksenov A.V., Druzhkov K.P., Kaptsov O.V., “Application of invariants of characteristics to construction of solutions without gradient catastrophe”, Int. J. Non-Linear Mech., 147 (2022), 104249, 1–8 | DOI

[26] Aksenov A.V., Druzhkov K.P., “Postroenie tochnykh reshenii sistemy uravnenii odnomernoi gazovoi dinamiki bez gradientnoi katastrofy”, Izv. RAN. Mekhan. zhidkosti i gaza, 2023, no. 1, 135–143 | DOI | Zbl

[27] Rozhdestvenskii B.L., Yanenko N.N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, Gl. red. fiz.-mat. lit-ry, M., 1978 | MR

[28] Aksenov A.V., “Vysshie invarianty kharakteristik uravneniya odnomernoi gazovoi dinamiki v lagranzhevykh peremennykh”, Mezhdunar. nauch. konf. “Matematika v sozvezdii nauk”. K yubileyu rektora MGU akad. V.A. Sadovnichego, Tez. dokl. (Moskva, 1–2 aprelya 2024), M., 2024, 101–103