Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Conservation laws and symmetrization of the equations of incompressible inviscid fluids

Author: Lubin G. Vulkov
Journal: Quart. Appl. Math. 57 (1999), 549-560
MSC: Primary 35L65; Secondary 76B99
DOI: https://doi.org/10.1090/qam/1704427
MathSciNet review: MR1704427
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Many physical processes are described by conservative quasilinear systems of equations that admit a supplementary independent conservation law. A typical example is the equations of incompressible fluid dynamics. If for basic equations the conservation laws of the momentum and the volume are chosen then the conservation law of the energy is an additional conservation law. In the present paper all conservation laws for incompressible fluid dynamics are derived and the symmetric forms of its equations are obtained. Godunov's method of symmetrization is generalized for the case of an arbitrary quasilinear system of equations.

References [Enhancements On Off] (What's this?)

  • [1] L. I. Judovič, Nonstationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz. 3, No. 6, 1032-1066 (1963); English transl. in USSR Comput. Math. and Math. Phys. 3 (1963)
  • [2] Roger Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis 20 (1975), no. 1, 32–43. MR 0430568
  • [3] S. N. Antoncev, A. V. Kažihov, and V. M. Monahov, Boundary Value Problems of Nonhomogeneous Fluid Mechanics, Nauka, Novosibirsk, 1983; English transl., North-Holland, Amsterdam, 1990
  • [4] R. Peyret and T. O. Taylor, Computational Methods in Fluid Flow, Springer, New York, 1982
  • [5] Peter J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734
  • [6] L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Y. Chapovsky; Translation edited by William F. Ames. MR 668703
  • [7] Nail H. Ibragimov, Transformation groups applied to mathematical physics, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1985. Translated from the Russian. MR 785566
  • [8] P. R. Eiseman and A. P. Stone, Conservation laws of fluid dynamics—a survey, SIAM Rev. 22 (1980), no. 1, 12–27. MR 554708, https://doi.org/10.1137/1022002
  • [9] B. L. Roždestvenskiĭ and N. N. Janenko, Systems of quasilinear equations and their applications to gas dynamics, Translations of Mathematical Monographs, vol. 55, American Mathematical Society, Providence, RI, 1983. Translated from the second Russian edition by J. R. Schulenberger. MR 694243
  • [10] D. Serre, Les invariants du premier ordre de l’equation d’Euler en dimension trois, Phys. D 13 (1984), no. 1-2, 105–136 (French, with English summary). MR 775281, https://doi.org/10.1016/0167-2789(84)90273-2
  • [11] E. D. Terent′ev and Ju. D. Šmyglevskiĭ, A complete system of equations in divergence form for the electromagnetic dynamics of a perfect gas, Ž. Vyčisl. Mat. i Mat. Fiz. 16 (1976), no. 3, 725–737, 821 (Russian). MR 0452108
  • [12] Lubin G. Vulkov, On the conservation laws in magnetohydrodynamics, Appl. Anal. 75 (2000), no. 1-2, 1–18. MR 1784808, https://doi.org/10.1080/00036810008840831
  • [13] L. G. Volkov, Conservation laws of gas dynamics in Lagrangian variables, Zh. Vychisl. Mat. i Mat. Fiz. 31 (1991), no. 10, 1552–1562 (Russian); English transl., Comput. Math. Math. Phys. 31 (1991), no. 10, 93–101 (1992). MR 1145224
  • [14] L. G. Vulkov, Conservation laws in nonlinear elasticity. I. One-dimensional elastodynamics, Quart. Appl. Math. 52 (1994), no. 3, 427–438. MR 1292195, https://doi.org/10.1090/qam/1292195
  • [15] Lubin G. Vulkov, On the conservation laws of the compressible Euler equations, Appl. Anal. 64 (1997), no. 3-4, 255–271. MR 1460082, https://doi.org/10.1080/00036819708840534
  • [16] S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521–523 (Russian). MR 0131653
  • [17] Amiram Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys. 49 (1983), no. 1, 151–164. MR 694161, https://doi.org/10.1016/0021-9991(83)90118-3
  • [18] S. K. Godunov, Symmetric form of the magnetohydrodynamic equations, Čisl. Metody Meh. Šplošnoi Sredy 3, No. 1, 26-34 (1972) (in Russian)
  • [19] A. A. Samarskiĭ and Yu. P. Popov, \cyr Raznostnye metody resheniya zadach gazovoĭ dinamiki, 2nd ed., “Nauka”, Moscow, 1980 (Russian). MR 619540
  • [20] Yu. I. Shokin, On conservatism of difference schemes of gas dynamics, Lecture Notes in Physics, vol. 264, 1986, pp. 578-583
  • [21] L. G. Volkov, On the construction of completely conservative difference schemes for equations in the nonlinear theory of elasticity, Zh. Vychisl. Mat. i Mat. Fiz. 31 (1991), no. 9, 1392–1401 (Russian); English transl., Comput. Math. Math. Phys. 31 (1991), no. 9, 92–99 (1992). MR 1145208

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35L65, 76B99

Retrieve articles in all journals with MSC: 35L65, 76B99

Additional Information

DOI: https://doi.org/10.1090/qam/1704427
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society