Geodesic
Conservation laws of shallow water theory and the Galilean relativity principle
Sibirskij žurnal industrialʹnoj matematiki, Tome 17 (2014) no. 1, pp. 99-113.

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Using an example of a model of shallow water theory, we show that the compatibility analysis of Hugoniot conditions for various basic systems of conservation laws in a coordinate system moving together with a strong discontinuity can lead to erroneous results. This is connected with the hierarchy of conservation laws of shallow water theory under a Galilean transformation, by which the law of conservation of the total energy is unconditionally noninvariant under this transformation. As a result, the Hugoniot condition corresponding to this conservation law depends on movement speed of the inertial coordinate system. It is shown that the specified defect of the classical model of shallow water theory is absent in the vortex shallow water model suggested by V. M. Teshukov.
Keywords: conservation laws of shallow water theory, Galilean relativity principle, classical and weak solutions. 
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V. V. Ostapenko. Conservation laws of shallow water theory and the Galilean relativity principle. Sibirskij žurnal industrialʹnoj matematiki, Tome 17 (2014) no. 1, pp. 99-113. http://geodesic.mathdoc.fr/item/SJIM_2014_17_1_a10/

[1] Lax P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Soc. Industr. Appl. Math., Philadelphia, 1972 | MR | Zbl

[2] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR

[3] Ostapenko V. V., Giperbolicheskie sistemy zakonov sokhraneniya i ikh prilozhenie k teorii melkoi vody, Izd-vo NGU, Novosibirsk, 2004

[4] Landau L. D., Lifshits E. M., Gidrodinamika, Nauka, M., 1988

[5] Stoker Dzh. Dzh., Volny na vode. Matematicheskaya teoriya i prilozheniya, Izd-vo inostr. lit., M., 1959

[6] Friedrichs K. O., “On the derivation of shallow water treory”, Comm. Appl. Math., 1 (1948), 109–134 | DOI | MR | Zbl

[7] Godunov S. K., “Interesnyi klass kvazilineinykh sistem”, Dokl. AN SSSR, 139:3 (1961), 521–523 | MR | Zbl

[8] Shugrin S. M., “Obodnom klasse kvazilineinykh sistem”, Dinamika sploshnoi sredy, 2, 1969, 145–148

[9] Friedrichs K. O., Lax P. D., “Systems of conservation equation with convex extension”, Proc. Nat. Acad. Sci. USA, 68:8 (1971), 1686–1688 | DOI | MR | Zbl

[10] Dressler R. F., “Comparison of theories and experiments for the hydraulic dam-break wave”, Intern. Assoc. Sci. Hydrology, 3:38 (1954), 319–328

[11] Stansby P. K., Chegini A., Barnes T. C., “The initial stages of dam-break flow”, J. Fluid Mech., 374 (1998), 407–424 | DOI | MR | Zbl

[12] Bukreev V. I., Gusev A. V., Malysheva A. A., Malysheva I. A., “Eksperimentalnaya proverka gazogidravlicheskoi analogii na primere zadachi o razrushenii plotiny”, Izv. RAN. MZhG, 2004, no. 5, 143–152 | Zbl

[13] Ostapenko V. V., “Techeniya, voznikayuschie pri razrushenii plotiny nad stupenkoi dna”, Prikl. mekhanika i tekhn. fizika, 44:4 (2003), 51–63 | MR | Zbl

[14] Ostapenko V. V., “Techeniya, voznikayuschie pri razrushenii plotiny nad ustupom dna”, Prikl. mekhanika i tekhn. fizika, 44:6 (2003), 107–122 | MR | Zbl

[15] Ostapenko V. V., “Techeniya, voznikayuschie pri razrushenii plotiny na skachke ploschadi secheniya pryamougolnogo kanala”, Prikl. mekhanika i tekhn. fizika, 53:5 (2012), 55–66 | MR

[16] Bukreev V. I., Gusev A. V., Ostapenko V. V., “Raspad razryva svobodnoi poverkhnosti zhidkosti nad ustupom dna kanala”, Izv. RAN. MZhG, 2003, no. 6, 72–83 | Zbl

[17] Bukreev V. I., Gusev A. V., Ostapenko V. V., “Volny v otkrytom kanale, obrazuyuschiesya pri udalenii schita pered nerovnym dnom tipa shelfa”, Vodnye resursy, 31:5 (2004), 1–6

[18] Kovyrkina O. A., Ostapenko V. V., “Sravnenie teorii i chislennogo eksperimenta v zadache o razrushenii plotiny na skachke ploschadi secheniya pryamougolnogo kanala”, Izv. RAN. MZhG, 2013, no. 3, 12–23 | Zbl

[19] Teshukov V. M., “Gazodinamicheskaya analogiya dlya vikhrevykh techenii so svobodnoi granitsei”, Prikl. mekhanika i tekhn. fizika, 48:3 (2007), 8–15 | MR | Zbl

[20] Richard G. L., Gavrilyuk S. L., “A new model of roll waves: comparison with Brock's experiments”, J. Fluid Mech., 698 (2012), 374–405 | DOI | MR | Zbl

[21] Brock R. R., “Development of roll-wave trains in open channels”, J. Hydraul. Div. ASCE, 95 (1969), 1401–1427

[22] Brock R. R., “Periodic permanent roll-waves”, J. Hydraul. Div. ASCE, 96 (1969), 2565–2580