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 Free Access

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.

**[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]**R. Temam,*On the Euler equations of an incompressible perfect fluid*, J. Functional Analysis**20**, No. 1, 32-43 (1975) 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]**P. J. Olver,*Applications of Lie Groups to Differential Equations*, Springer, New York, 1986 MR**836734****[6]**L. V. Ovsiannikov,*Group Analysis of Differential Equations*, Academic Press, New York, 1982 MR**668703****[7]**N. H. Ibragimov,*Transformation Groups Applied to Mathematical Physics*, Reidel, Boston, 1985 MR**785566****[8]**P. R. Eiseman and A. P. Stone,*Conservation laws of fluid dynamics, a survey*, SIAM Review**22**12-27 (1980) MR**554708****[9]**B. L. Roždestvenskii and N. N. Janenko,*Systems of Quasilinear Equations and their Applications to Gas Dynamics*, Nauka, Moscow, 1968; English transl. of 2nd ed., Amer. Math. Soc., Providence, RI, 1983 MR**694243****[10]**D. Serre,*Les invariants du premier ordre de l'équation d'Euler en dimension trois*, Phys. D**13**, 105-136 (1984) MR**775281****[11]**D. Terentev and J. D. Šmyglevskii,*A complete set of equations in divergence form for the electromagnetic dynamics of a perfect gas*, Ž. Vyčisl. Mat. i Mat. Fiz.**25**, No. 3, 725-737 (1976); English transl. in USSR Comput. Math. and Math. Phys.**16**(1976) MR**0452108****[12]**L. G. Vulkov,*On the conservation laws in magnetohydrodynamics at infinite conductivity*, Ž. Vyčisl.**25**, No. 7, 1401-1408 (1985); English transl. in USSR Comput. Math. and Math. Phys.**25**(1985) MR**1784808****[13]**L. G. Vulkov,*Conservation laws of the two-dimensional gas dynamics equations in Lagrange variables*, Ž. Vyčisl. Mat. i Mat. Fiz.**3**, No. 10, 1552-1562 (1991); English transl. in USSR Comput. Math. and Math. Phys.**31**(1991) MR**1145224****[14]**L. G. Vulkov,*Conservation laws in nonlinear elasticity*, I.*One-dimensional elastodynamics*, Quart. Appl. Math.**52**427-438 (1994) MR**1292195****[15]**L. G. Vulkov,*On the conservation laws of the compressible Euler equations*, Applicable Analysis**64**, No. 3-4, 255-271 (1997) MR**1460082****[16]**S. K. Godunov,*An interesting class of quasilinear systems*, DAN SSSR**139**, No. 3, 521-523 (1961); English transl. in Soviet Math. Dokl.**2**(1961) MR**0131653****[17]**A. Harten,*On the symmetric form of systems of conservation laws with entropy*, J. Comp. Phys.**49**, 151-169 (1989) MR**694161****[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. Samarskii and Yu. P. Popov,*Difference methods for the solution of problems of gas dynamics*, M: Nauka, 1980 (in 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. Vulkov,*On construction of fully conservative difference schemes of the nonlinear elasticity equations*, Ž. Vyčisl. Mat. i Mat. Fiz.**31**, No. 9, 1392-1401 (1991); English transl. in USSR Comput. Math. and Math. Phys.**31**(1991) MR**1145208**

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