In this article we investigate the Cauchy problem for a system of nonlinear integro-differential equations of gas dynamics that describes the motion of a finite mass of a self-gravitating gas bounded by a free boundary. It is assumed that gas moving is considered under the condition that at any time the free boundary consists of the same particles. This makes convenient the transition from Euler to Lagrangian coordinates. Initially, this system in Euler coordinates is transformed into a system of integro-differential equations in Lagrangian coordinates. A lemma on equivalence of these systems is proved. Then the system in Lagrange variables is transformed into a system consisting of Volterra integral equations and the equation continuity, for which the existence theorem for the solution of the Cauchy problem is proved with the help of the method of successive approximations. Based on the mathematical induction, the continuity of the solution and belonging of the solution to the space of infinitely differentiable functions are proved. The boundedness and the uniqueness of the solution are proved. The solution of a system of Volterra integral equations defines the mapping of the initial domain into the domain of moving gas, and also sets the law of motion of a free boundary, as a mapping of points of an initial boundary.

Nikolai Chuev, Cand. Sci. (Phys.–Math.), Assoc. Prof., Ural State University of Railway Transport, 66, Kolmogorov st., Yekaterinburg, 620034, Russian Federation, tel.: (343)2212444, email: n_chuev44@mail.ru

Chuev N.P. The Cauchy Problem for System of Volterra Integral Equations Describing the Motion of a Finite Mass of a Self-gravitating Gas. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 33, pp. 35-50. (in Russian) https://doi.org/10.26516/1997-7670.2020.33.35

1. Andreev V.K. Ustoychivost’ neustanovivshikhsya dvizheniy zhidkosti so svobodnoy granitsey [Stability of unsteady fluid motions with a free boundary]. Novosibirsk, Nauka Publ., 1992, 136 p. (in Russian)
2. Bogoiavlenskii O.I. Dynamics of a gravitating gaseous ellipsoid. J. Appl Math. Mech., 1976, no. 40, pp. 246-256.
3. Vasilieva A.B., Tikhonov N.A. Integral’nye uravneniya [Integral equations]. Мoscow, Fizmatlit Publ., 2002, 160 p. (in Russian)
4. Gunter N.M. Teoriya potentsiala i ee primenenie k osnovnym zadacham matematicheskoy fiziki [Potential theory and its application to the basic problems of mathematical physics]. Мoscow, Gostehizdat Publ., 1953, 415 p. (in Russian)
5. Kartashev A.P., Rozhdestvenskii B.L. Obyknovennye differentsial’nye uravneniya i osnovy variatsionnogo ischisleniya [Ordinary differential equations and fundamentals of the calculus of variations]. Мoscow, Nauka Publ., 1979, 228 p. (in Russian)
6. Lamb H. Gidrodinamika [Hydrodynamics]. Мoscow, OGIZ Publ., 1947, 929 p.(in Russian)
7. Nikolsky S.М. Kurs matematicheskogo analiza, Т. II [A Course in Mathematical Analysis, vol. II.] Мoscow, Nauka Publ., 1983, 448 p. (in Russian)
8. Ovsyannikov L.V. Lektsii po osnovam gazovoy dinamiki [Lectures on the basics of gas dynamics]. Мoscow-Izhevsk, Institut kompyuternih issledovanii Publ., 2003, 336 p. (in Russian)
9. Ovsyannikov L.V. Obshchie uravneniya i primery. Zadacha o neustanovivshemsya dvizhenii zhidkosti so svobodnoy granitsey [General equations and examples]. Novosibirsk, Nauka Publ., 1967, 75 p. (in Russian)
10. Smirnov N.S. Vvedenie v teoriyu nelineynykh integral’nykh uravneniy [Introduction to the theory of nonlinear integral equations]. Moscow, Leningrad, 1936, 125 p. (in Russian)
11. Sretensky L.N. Teoriya n’yutonovskogo potentsiala [Theory of Newtonian potential]. Moscow, OGIZ Publ., 1946, 318 p. (in Russian)
12. Stanyukovich K.P. Neustanovivshiesya dvizheniya sploshnoy sredy [Unsteady motion of continuum media]. Moscow, Nauka Publ., 1971, 875 p. (in Russian)
13. Chandrasekhar S. Ellipsoidal’nye figury ravnovesiya [Ellipsoidal figures of equilibrium] Мoscow, Mir Publ., 1973, 228 p.
14. Chuev N.P. On the existence and uniqueness of a solution to a problem Cauchy for a system of integral equations describing the motion of a rarefied mass of a selfgravitating gas. Journal of Computational Mathematics and Mathematical Physics, 2020, vol. 60, no. 4, pp. 663-672. https://doi.org/10.1134/S0965542520040077
15. Hartman F. Ordinary Differential Equations. Moscow, Mir Publ., 1970, 720 p. (in Russian)
16. Evans L.C. Partial differential equations. Novosibirsk, 2003, 562 p. (in Russian)