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Conservation laws for plane steady potential barotropic flow

Published online by Cambridge University Press:  03 June 2013

YU. A. CHIRKUNOV  and
S. B. MEDVEDEV
YU. A. CHIRKUNOV
Affiliation:
Institute of Computational Technologies SB RAS, Novosibirsk, Russia Department of Applied Mathematics and Computer Science, Novosibirsk State Technical University, Novosibirsk, Russia email: [email protected]
S. B. MEDVEDEV
Affiliation:
Institute of Computational Technologies SB RAS, Novosibirsk, Russia Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia email: [email protected]
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Abstract

It is shown that the set of conservation laws for the nonlinear system of equations describing plane steady potential barotropic flow of gas is given by the set of conservation laws for the linear Chaplygin system. All the conservation laws of zero order for the Chaplygin system are found. These include both known and new nonlinear conservation laws. It is found that the number of conservation laws of the first order is not more than three, assuming that the laws do not depend on the velocity potential and are not non-obvious ones. The components of these conservation laws are quadratic with respect to the stream function and its derivatives. All the Chaplygin functions are found, for which the Chaplygin system has three non-obvious conservation laws of the first order that are independent of velocity potential. All such non-obvious first-order conservation laws are found.

Keywords

Conservation lawsSteady barotropic flowChaplygin equationChaplygin system
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 6 , December 2013 , pp. 789 - 801
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Chen, G.-Q., Slemrod, M. & Wang, D. (2008) Vanishing viscosity method for transonic flow. Arch. Ration. Mech. Anal. 189, 159–188.Google Scholar
[2]Chirkunov, Yu. A. (2007) Group Analysis of Linearand Quasi-Linear Differential Equations, NGUEU, Novosibirsk, Russia (in Russian).Google Scholar
[3]Chirkunov, Yu. A. (2010) On the symmetry classification and conservation laws for quasilinear differential equations of second order. Math. Notes 87, 115–121.CrossRefGoogle Scholar
[4]Ibragimov, N. H. (1994) CRC Handbook of Lie Group Analysis of Differential Equations, Vol. I: Symmetries, Exact Solutions, and Conservation Laws, CRC Press, Boca Raton, FL.Google Scholar
[5]Ibragimov, N. Kh. (2004) Invariants of hyperbolic equations: Solution of the Laplace problem. J. Appl. Mech. Tech. Phys. 2, 158–166.Google Scholar
[6]Laplace, P. S. (1893) Recherches sur le calcul integral aux differences partielles. Memoires de Sciences de Paris. 1773/77, pp. 341–402. (English translation: Laplace, P. S. (1966) Oeuvres Completes, Vol. 9, Gauthier-Villars, Paris, France, pp. 5–68.Google Scholar
[7]Loewner, C. (1953) Conservation laws in compressible flow and associated mappings. J. Ration. Mech. Anal. 2, 537–561.Google Scholar
[8]Mises von, R. (1958) Mathemetical Theory of Compressible Fluid Flow, Academic Press, New York.Google Scholar
[9]Morawetz, C. S. (1985) On a weak solution for a transonic flow problem. Comm. Pure Appl. Math. 38, 797–818.Google Scholar
[10]Olver, P. (1986) Applications of Lie Groups to Differential Equations, Springer-Verlag, New York.Google Scholar
[11]Ovsyannikov, L. V. (1960) Group properties of the Chaplygin equation. J. Appl. Mech. and Tech. Phys. 3, 126–145 (in Russian). [English translation in: Lie Group Analysis: Classical Heritage, ALGA, Karlskrona, Sweden (2004)].Google Scholar
[12]Ovsyannikov, L. V. (1962) Group Properties of Differential Equations, SB AS, Novosibirsk, Russia (in Russian).Google Scholar
[13]Ovsyannikov, L. V. (1981) Lectures on the Fundamentals of Gas Dynamics, Nauka, Moscow (in Russian).Google Scholar
[14]Ovsyannikov, L. V. (1982) Group Analysis of Differential Equations. New York, Academic press.Google Scholar
[15]Rylov, A. I. (2002) Equations SA Chaplygin and an infinite number of uniformly-divergent equations of gas dynamics. Doklady RAN 383, 34–36.Google Scholar