Géométrie des systèmes hyperboliques de lois de conservation
Mémoires de la Société Mathématique de France, no. 56 (1994) , 132 p.
@book{MSMF_1994_2_56__1_0,
     author = {Sevennec, Bruno},
     title = {G\'eom\'etrie des syst\`emes hyperboliques de lois de conservation},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {56},
     year = {1994},
     doi = {10.24033/msmf.370},
     mrnumber = {95g:35123},
     zbl = {0807.35090},
     language = {fr},
     url = {http://archive.numdam.org/item/MSMF_1994_2_56__1_0/}
}
TY  - BOOK
AU  - Sevennec, Bruno
TI  - Géométrie des systèmes hyperboliques de lois de conservation
T3  - Mémoires de la Société Mathématique de France
PY  - 1994
IS  - 56
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/MSMF_1994_2_56__1_0/
DO  - 10.24033/msmf.370
LA  - fr
ID  - MSMF_1994_2_56__1_0
ER  - 
%0 Book
%A Sevennec, Bruno
%T Géométrie des systèmes hyperboliques de lois de conservation
%S Mémoires de la Société Mathématique de France
%D 1994
%N 56
%I Société mathématique de France
%U http://archive.numdam.org/item/MSMF_1994_2_56__1_0/
%R 10.24033/msmf.370
%G fr
%F MSMF_1994_2_56__1_0
Sevennec, Bruno. Géométrie des systèmes hyperboliques de lois de conservation. Mémoires de la Société Mathématique de France, Serie 2, no. 56 (1994), 132 p. doi : 10.24033/msmf.370. http://numdam.org/item/MSMF_1994_2_56__1_0/

[A1] V.I. Arnold, Méthodes mathématiques de la mécanique classique, Mir 1976. | Zbl | MR

[A2] V.I. Arnold, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir 1978. | Zbl

[Ba] C. Bardos, Apparition éventuelle de singularités dans les problèmes d'évolution non linéaires, séminaire Bourbaki No 555 (1980). | Zbl | Numdam

[Bla] W. Blaschke, Vorlesungen über Differentialgeometrie ; II, Affine Differentialgeometrie, Berlin, J. Springer, 1923, cf. aussi Gesammelte Werke, Band IV (Burau, Chern et al editors), Thales, Essen, 1985.

[Bo1] G. Boillat, Chocs caractéristiques, C.R.A.S. 274, sér. A, 1018-1021 (1972). | Zbl | MR

[Bo2] G. Boillat, Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systèmes hyperboliques, C.R.A.S. 278, sér. A, 909-912 (1974). | Zbl | MR

[Bo3] G. Boillat, Symétrisation des systèmes d'équations aux dérivées partielles avec densité d'énergie convexe et contraintes, C.R.A.S. 295, sér. 1, 551-554 (1982). | Zbl | MR

[B-H] M. Brio, J.K. Hunter, Rotationnaly invariant hyperbolic waves, C.P.A.M. 43 (1990), 1037-1053. | Zbl | MR

[Chk] A.V. Chakmazyan, Normal connection in the geometry of normalized submanifolds of affine space, Soviet Math. (Plenum Publ. Corp.), 2131-2140 (1991). | Zbl

[Co-D] B.D. Coleman, E.H. Dill, Z. Angew. Math. Phys. 22, 691-702 (1971). | Zbl

[Co-S1] C.C. Conley, J.A. Smoller, Shock waves as limits of progressive wave solutions of higher order equations, C.P.A.M. 24, 459-472 (1971), Part II : C.P.A.M. 25, 133-146 (1972). | Zbl | MR

[Co-S2] C.C. Conley, J.A. Smoller, On the structure of magnetohydrodynamic shock waves, C.P.A.M. 27, 367-375 (1974). | Zbl | MR

[Daf1] C.M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Diff. Eqs 20, 90-114 (1976). | Zbl | MR

[Daf2] C.M. Dafermos, Hyperbolic systems of conservation laws, “Systems of nonlinear partial differential equations”, J. Ball (ed.), NATO ASI series C, No. 111, Reidel Publ. Co., 25-70 (1983). | Zbl | MR

[Daf3] C.M. Dafermos, Quasilinear hyperbolic systems with involutions, Arch. Rat. Mech. Anal., 94, 373-389 (1986). | Zbl | MR

[Daf4] C.M. Dafermos, Admissible wave fans in nonlinear hyperbolic systems, Arch. Rat. Mech. Anal., 106:3, 243-260 (1989). | Zbl | MR

[Daf5] C.M. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal., 107:2, 127-157 (1989). | Zbl | MR

[Dar1] G. Darboux, Leçons sur la théorie générale des surfaces. Partie IV, Gauthier-Villars 1946. | Zbl

[Dar2] G. Darboux, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars 1910. | JFM

[Di-V] F. Dillen, L. Vrancken, Affine differential geometry of hypersurfaces, Geometry and Topology of Submanifolds II, Avignon 1988 (World Scientific 1990), 144-164. | Zbl | MR

[DiP1] R.J. Diperna, Global solutions to a class of nonlinear hyperbolic systems of equations, C.P.A.M. 26, 1-28 (1973). | Zbl | MR

[DiP2] R.J. Diperna, Existence in the large for quasilinear hyperbolic conservation laws, Arch. Rat. Mech. Anal. 52, 244-257 (1973). | Zbl | MR

[DiP3] R.J. Diperna, Decay of solutions of hyperbolic systems of conservation laws with a convex extension, Arch. Rat. Mech. Anal. 64, 1-46 (1977). | Zbl | MR

[DiP4] R.J. Diperna, Uniqueness of solutions of nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J. 28, 137-188 (1979). | Zbl | MR

[DiP5] R.J. Diperna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. 82, 27-70 (1983). | Zbl | MR

[DiP6] R.J. Diperna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 88, 223-270 (1985). | Zbl | MR

[DiP-M] R.J. Diperna, A. Majda, The validity of geometrical optics for weak solutions of conservation laws, Comm. Math. Phys. 98, 313-347 (1985). | Zbl | MR

[Du-N1] B.A. Dubrovin, S.P. Novikov, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Russian Math. Surveys 44:6, 35-124 (1989). | Zbl | MR

[Du-N2] B.A. Dubrovin, S.P. Novikov, Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method, Soviet Math. Dokl. Vol 27, no3, 665-669 (1983). | Zbl | MR

[Dy] V.F. D'Yacenko, Cauchy's problem for quasilinear sytems, Sov. Math. Dokl. 2, 7-8 (1961). | Zbl

[Fe1] E.V. Ferapontov, Hamiltonian systems of hydrodynamic type and their realization on hypersurfaces of a pseudo-euclidean space, Journal of Sov. Math., p. 1970-1995, Plenum (1991). | Zbl | MR

[Fe2] E.V. Ferapontov, Integration of weakly nonlinear hydrodynamic systems in Riemann invariants, Physics Letters A, p. 112-118, (1991). | MR

[Fe-P] E.V. Ferapontov, M.V. Pavlov, Quasiclassical limit of coupled KdV equations. Riemann invariants and multihamiltonian structure. Physica D, 52, (1991) 211-219. | Zbl | MR

[Fi-G] P.C. Fife, X. Geng, Mathematical aspects of electrophoresis, Proc. Heriot-Watt 1988 : Reaction-Diffusion Equations, Oxford Univ. Press.

[Fre1] H. Freistühler, Anomale Schocks, strukturell labile Lösungen und die Geometrie der Rankine-Hugoniot Bedingungen, Thèse, Ruhr-Universität Bochum (1987).

[Fre2] H. Freistühler, On compact linear degeneracy, IMA preprint 551, University of Minnesota, (1989).

[Fre3] H. Freistühler, Rotationnal degeneracy of hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal. 113, 1991, 39-64. | Zbl

[Fri] K.O. Friedrichs, On the laws of relativistic electro-magneto-fluid dynamics, C.P.A.M. 27, 749-808 (1974). | Zbl | MR

[Fri-L] K.O. Friedrichs, P.D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. USA 68, 1686-1688 (1971). | Zbl | MR

[F-R-S] S. Friedland, J.W. Robbin, J.H. Sylvester, On the crossing rule, C.P.A.M. 37, 19-37 (1984). | Zbl | MR

[Ga] C.S. Gardner, Korteweg-de Vries equation and generalizations. IV : the Korteweg-de Vries equation as a Hamiltonian system, J. Math. Phys 12 (1971), 1548-1551. | Zbl | MR

[Gel] I.M. Gel'Fand, Some problems in the theory of quasilinear equations, A.M.S. Trans. Ser. 2, No 29, 295-381 (1963). | Zbl | MR

[Ger] P. Germain, Contribution à la théorie des ondes de choc en magnétodynamique des fluides, ONERA Publ. No 97, Paris 1959. | MR

[Gl] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, C.P.A.M. 18, 697-715 (1965). | Zbl | MR

[Gl-L] J. Glimm, P.D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. A.M.S. No 101, Providence 1970. | Zbl | MR

[Go1] S.K. Godunov, On the concept of generalized solution, Sov. Math. Dokl. 1, 1194-1196 (1960). | Zbl | MR

[Go2] S.K. Godunov, On non-unique "blurring" of discontinuities in solutions of quasilinear systems, Sov. Math. Dokl. 2, 43-44 (1961). | Zbl | MR

[Go3] S.K. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl. 2, 947-949 (1961). | Zbl

[Go4] S.K. Godunov, Lois de conservation et intégrales d'énergie des équations hyperboliques, Proc. 1986 Nonlinear Hyperbolic Problems XV, p. 135-149, C. Carasso, P.A. Raviart, D. Serre eds., Lect. Notes Math. 1270, Springer-Verlag 1987. | Zbl

[Gue] O. Guès, Problèmes mixtes hyperboliques quasilinéaires caractéristiques, Thèse, Université de Rennes-1, (1989).

[Gui-St] V. Guillemin, S. Sternberg, Symplectic techniques in physics, Cambridge Univ. Press, 1984. | Zbl | MR

[Ha-L-vL] A. Harten, P.D. Lax, B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic systems of conservation laws, SIAM Review, 25, 1, 35- (1983). | Zbl | MR

[Hb1] A. Heibig, Etude variationnelle du problème de Riemann, Thèse, Lyon 1989.

[Hb2] A. Heibig, Régularité des solutions du problème de Riemann, Comm. in P.D.E., 15, 5, 693-709 (1990). | Zbl | MR

[Hb3] A. Heibig, Existence et unicité des solutions pour certains systèmes de lois de conservation, Prepubl. ENS-Lyon no 32, (1990) (à paraître dans Arch. Rat. Mech. Anal.). | MR

[Hb4] A. Heibig, Error estimates for oscillating solutions to hyperbolic systems of conservation laws, Comm. in P.D.E., 18, 2, 281-304 (1993). | Zbl | MR

[Hel] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Acad. Press, New York, 1978. | Zbl

[H-M-R] J.K. Hunter, A. Majda, R. Rosales, Resonantly interacting weakly nonlinear waves. II. Several space variables, Stud. Appl. Math. 75, 187-226 (1986). | Zbl | MR

[Je] A. Jeffrey, Quasilinear hyperbolic systems and waves, Pitman, London 1976. | Zbl | MR

[Jn] F. John, Formation of singularities in one-dimensional nonlinear wave propagation, C.P.A.M., Vol 27, 377-405 (1974). | Zbl | MR

[Jo-M-R1] J.L. Joly, G. Metivier, J. Rauch, Rigorous resonant 1-d nonlinear geometric optics, Journées EDP, St-Jean de Monts, 1990. | Numdam | Zbl | MR | EuDML

[Jo-M-R2] J.L. Joly, G. Metivier, J. Rauch, Formal and rigorous nonlinear high frequency waves, Proc. Nonlin. Hyp. Eq., Como 1990, Pitman. (à paraître)

[Jou] E. Jouguet, Sur la propagation des discontinuités dans les fluides, C.R.A.S. 132, 673-676 (1901). | JFM

[K-K1] B.L. Keyfitz, H.C. Kranzer, The Riemann problem for some nonstrictly hyperbolic systems of conservation laws, Notices AMS, 23 (1976), A-127-128.

[K-K2] B.L. Keyfitz, H.C. Kranzer, Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two non-linear conservation laws, J. Diff. Equ., 27 (1978), 444-476. | Zbl | MR

[K-K3] B.L. Keyfitz, H.C. Kranzer, A system of non-stictly hyperbolic conservation laws arising in elasticity theory, Arch. Rat. Mech. Anal., 72 (1980) 219-241. | Zbl | MR

[K-N] S. Kobayashi, K. Nomizu, Foundations of differential geometry I,II, Wiley Interscience, New York, 1963,1969. | Zbl

[Kru] S.N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10, 217-242 (1970). | Zbl

[K-T] N.N. Kuznetsov, V.A. Tupshiev, A certain generalization of a theorem of Glimm, Dokl. Akad. Nauk. SSSR, 221, 287-290 (1975). | Zbl | MR

[Law] H.B. Lawson Jr, The quantitative theory of foliations, CBMS-NSF Reg. Conf. Ser. in Math. No 27, AMS, Providence, 1977. | Zbl | MR

[Lax1] P.D. Lax, The initial value problem for nonlinear hyperbolic equation in two independent variables, Ann. of Math. Studies 33, Princeton, 211-229 (1954). | Zbl | MR

[Lax2] P.D. Lax, Hyperbolic systems of conservation laws: II, CPAM Vol 10, 537-566 (1957). | Zbl | MR

[Lax3] P.D. Lax, Shock waves and entropy, Contributions to nonlinear functional analysis, Zarantonello ed., p. 603-634, NY Academic Press, 1971. | Zbl | MR

[Lax4] P.D. Lax, The formation and decay of shock waves, Amer. Math. Monthly 79, 227-241 (1972). | Zbl | MR

[Lax5] P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS-NSF Reg. Conf. Ser. in Appl. Math. no 11 SIAM (1973). | Zbl | MR

[Lax6] P.D. Lax, The multiplicity of eigenvalues, B.A.M.S. Vol 6 No 2, 213-214 (1982). | Zbl | MR

[Lax-Lv] P.D. Lax, C.D. Levermore, The small dispersion limit for the KdV equation I, II & III, C.P.A.M. 36 (1983), 253-290, 571-593, 809-830 | Zbl | MR

[Lev] C.D. Levermore, The hyperbolic nature of the zero dispersion KdV limit, Comm. Part. Diff. Equ., 13 (1988), 495-514. | Zbl | MR

[Li1] T.-P. Liu, Existence and uniqueness for Riemann problems, Trans. A.M.S., 212, 375-382 (1975). | Zbl | MR

[Li2] T.-P. Liu, The Riemann problem for general systems of conservation laws, J. Diff. Eqs., 18, 218-234 (1975). | Zbl | MR

[Li3] T.-P. Liu, Solutions in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J. 26,1, 147-177 (1977). | Zbl | MR

[Li4] T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57, 135-148 (1977). | Zbl | MR

[Li5] T.-P. Liu, Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations, J. Diff. Eqs, 33, 92-111 (1979). | Zbl | MR

[Li6] T.-P. Liu, Admissible solutions of hyperbolic conservation laws, Mem. A.M.S. No 240, Providence 1981. | Zbl | MR

[Ma-P] A. Majda, R.L. Pego, Stable viscosity matrices for systems of conservation laws, J. Diff. Eqs. 56, 229-262 (1985). | Zbl | MR

[Ma-R] A. Majda, R. Rosales, Resonantly interacting weakly nonlinear waves. I. A single space variable, Stud. Appl. Math. 71, 149-179 (1984). | Zbl | MR

[Ma-R-S] A. Majda, R. Rosales, M. Schonbek, A canonical system of integro-differential equations arising in resonant nonlinear acoustics, Stud. Appl. Math. 79, 205-262 (1988). | Zbl | MR

[Mo1] M.S. Mock, Systems of conservation laws of mixed type, J. Diff. Eqs. 37, 70-88 (1980). | Zbl | MR

[Mo2] M.S. Mock, A topological degree for orbits connecting critical points of autonomous systems, J. Diff. Eqs. 38, 176-191 (1980). | Zbl | MR

[Mo3] M. Sever, Existence in the large for Riemann problems for systems of conservation laws, Trans. A.M.S. 292, 375-381 (1985). | Zbl | MR

[Mo4] M. Sever, Uniqueness failure for entropy solutions of hyperbolic systems of conservation laws, C.P.A.M. 42, 173-183 (1989), erratum C.P.A.M. 43, 295-297 (1990). | Zbl | MR

[Mo5] M. Sever, The rate of total entropy generation for Riemann problems, J. Diff. Eqs. 87, 115-143 (1990). | Zbl | MR

[Mo-F] O.I. Mokhov, E.V. Ferapontov, Nonlocal hamiltonian operators of hydrodynamic type and constant curvature metric, Russ. Math. Surv. 45, 3 (1990) 218-219. | Zbl | MR

[Ni-Sm] T. Nishida, J.A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, C.P.A.M. 26, 183-200 (1973). | Zbl | MR

[No-Pi] K. Nomizu, U. Pinkall, Cubic form theorem for affine immersions, Results in Math. (Birkhäuser), 13, 338-362 (1988). | Zbl | MR

[Ole1] O.A. Oleinik, Cauchy's problem for nonlinear equations in a class of discontinuous functions, Dokl. 95 (1954), A.M.S. Transl. (2) 42, 7-12 (1964). | Zbl | MR

[Ole2] O.A. Oleinik, Discontinuous solutions of nonlinear differential equations, Uspekhi (1957), A.M.S. Transl. (2) 26, 95-172 (1963). | Zbl

[Ole3] O.A. Oleinik, Uniqueness and stability of the generalized solutions of the Cauchy problem for a quasilinear equation, Uspekhi (1959), A.M.S. Transl. (2) 33, 285-290 (1963). | Zbl

[Olv] P.J. Olver, Applications of Lie groups to differential equations, Springer Verlag, 1986. | Zbl | MR

[Ovs] L.V. Ovsiannikov, Group analysis of differential equations, Acad. Press, 1982. | Zbl | MR

[Peg] R. Pego, Some explicit resonating waves in weakly nonlinear gas dynamics, Stud. Appl. Math. 79, 263-270 (1988). | Zbl | MR

[Ran] W.J.M. Rankine, On the thermodynamical theory of waves of finite longitudinal disturbance, Phil. Trans. Roy. Soc. Lond. 160, 277-288 (1870). | JFM

[Rie] B. Riemann, Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Ges. Werke, Leipzig, 157-175 (1892).

[Sch] M. Schatzman, Continuous Glimm functionals and uniqueness of solutions of the Riemann problem, Ind. Univ.Math.J. 34, 533-589 (1985). | Zbl | MR

[Ser1] D. Serre, Systèmes hyperboliques non-linéaires commutant entre eux, Prepubl. Lyon-St Etienne No 26 (1984).

[Ser2] D. Serre, Compacité par compensation et systèmes hyperboliques de lois de conservation, C.R.Acad.Sc. Paris, t. 299, ser.I, No 20, p.555-558 (1984). | Zbl | MR

[Ser3] D. Serre, La compacité par compensation pour les systèmes hyperboliques non-linéaires de deux équations à une dimension d'espace, J.M.P.A. 65, 423-468 (1986). | Zbl | MR

[Ser4] D. Serre, Solutions à variation bornée pour certains systèmes hyperboliques de lois de conservation, J. of Diff. Equ. 68, 137-169 (1987). | Zbl | MR

[Ser5] D. Serre, Domaines invariants pour les systèmes hyperboliques de lois de conservation, J. of Diff. Equ. 69, 46-62 (1987). | Zbl | MR

[Ser6] D. Serre, Existence globale d'ondes planes en électromagnétisme non-linéaire, C.R.Acad.Sc. Paris, t. 304, ser.I, No 20 (1987). | Zbl | MR

[Ser7] D. Serre, Systèmes hyperboliques riches de lois de conservation, Pre-publ. Lyon-St Etienne no 74 (1988), et Sém. Collège de France, “non-linear PDEs and their applications”, H. Brézis & J.-L. Lions eds., Pitman (à paraître).

[Ser8] D. Serre, Les ondes planes en éléctromagnétisme non linéaire, Physica D, No 38, 227-251 (1988). | Zbl | MR

[Ser9] D. Serre, Richness and the classification of quasilinear hyperbolic systems, IMA vol. in Math. and their appl. Vol 29, Springer Verlag, 1991. | Zbl | MR

[Ser10] D. Serre, Systèmes d'EDO invariants sous l'action de systèmes hyperboliques d'EDP, Ann. Inst. Fourier, 39, 4, 953-968 (1989). | Zbl | MR | Numdam

[Ser11] D. Serre, Oscillations non-linéaires des systèmes hyperboliques : méthodes et résultats qualitatifs, Ann. Inst. Henri Poincaré, Analyse non linéaire, Vol 8, No 34, p. 351-417, 1991. | Zbl | MR | Numdam

[Ser12] D. Serre, Oscillations non-linéaires de haute fréquence ; dim=1, Pre-publ. Lyon-St Etienne no 97 (1990) et Sém. Collège de France, “non-linear PDEs and their applications”, H. Brézis & J.-L. Lions eds., Pitman (à paraître).

[Ser13] D. Serre, Quelques méthodes d'étude de la propagation d'oscillations hyperboliques non-linéaires, Séminaire EDP 1990-1991, Ecole Polytechnique, exposé No XX. | Zbl | Numdam

[Ser14] D. Serre, Un modèle relaxé pour les câbles inextensibles, MMAN,25,4 (1991) 465-481. | Zbl | MR | Numdam

[Ser15] D. Serre, Intégrabilité d'une classe de systèmes de lois de conservation, Prépubl. ENS-Lyon No 45 (1991), à paraître dans Forum Math. | Zbl

[Sm] J.A. Smoller, Shock waves and reaction-diffusion equations, Grund. Math. Wiss. 258, Springer Verlag 1983. | Zbl | MR

[Sp] M. Spivak, A comprehensive introduction to differential geometry, Tome III, Publish or Perish Inc., 1975. | Zbl

[Sto] G.G. Stokes, On a difficulty in the theory of sound, Philosophical Magazine 33, 349-356, (1848).

[Str1] J.W. Strutt (Lord Rayleigh), The Theory of Sound, Vol. II, London : McMillan 1878.

[Str2] J.W. Strutt (Lord Rayleigh), Note on tidal bores, Proc. Roy. Soc. London A 81,448-449 (1908). | JFM

[Te1] B. Temple, Solutions in the large for some nonlinear hyperbolic conservation laws of gas dynamics, J. Diff. Eqs. 41, 96-161 (1981). | Zbl

[Te2] B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280, 781-795 (1983). | Zbl | MR

[Ts] S.P. Tsarev, On the Liouville Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type arising in the Bogolyubov-Whitham averaging theory, Russian Math. Surveys 39 : 6, 227-228 (1984). | Zbl | MR

[Ts2] S.P. Tsarev, On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Soviet Math. Dokl. 31 No3, 488-491 (1985). | Zbl | MR

[Ts3] S.P. Tsarev, Ph. D. Thesis, Moscow state University, 1986.

[Ts4] S.P. Tsarev, The geometry of hamiltonian systems of hydrodynamic type. The generalized hodograph method, Math. USSR Izvsetiya 37 No2 (1991) 397-419. | Zbl | MR

[Vo] A.I. Vol'Pert, The BV space and quasilinear equations, Math. USSR Sb. 2, 225-267 (1967). | Zbl

[Vv] N.D. V'Vedenskaya, An example of non-uniqueness of a generalized solution of a quasilinear system of equations, Sov. Math. Dokl. 2, 89-90 (1961). | Zbl

[Wa] D. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J.Diff.Eqs. 68, 118-136 (1987). | Zbl | MR

[Wh] G.B. Whitham, Linear and nonlinear waves, Wiley NY 1974. | Zbl

Cited by Sources: