Geodesic
Hyperbolicity of a class of first-order quasilinear covariant equations of divergent type
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Winter Mathematical School "Modern Methods of Function Theory and Related Problems", Voronezh, January 28 - February 2, 2021, Part 2, Tome 207 (2022), pp. 16-26.

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A special class of systems of first-order quasilinear partial differential equations is considered. These divergent-type systems are invariant under time and space translations; they are transformed covariantly under the action of the rotation group. We give a description of the class of nonlinear first-order differential operators corresponding to the systems of the considered class and prove a theorem on the equivalence of the concepts of hyperbolicity and hyperbolicity in the sense of Friedrichs.
Keywords: first-order differential operator, quasilinear system, hyperbolicity, vector field, covariance, field flux density, symmetric tensor.
Mots-clés : divergence 
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Yu. P. Virchenko; A. E. Novoseltseva. Hyperbolicity of a class of first-order quasilinear covariant equations of divergent type. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Winter Mathematical School "Modern Methods of Function Theory and Related Problems", Voronezh, January 28 - February 2, 2021, Part 2, Tome 207 (2022), pp. 16-26. http://geodesic.mathdoc.fr/item/INTO_2022_207_a2/

[4] Virchenko Yu. P., Subbotin A. V., “Matematicheskie zadachi konstruirovaniya evolyutsionnykh uravnenii dinamiki kondensirovannykh sred”, Mat. Mezhdunar. nauch. konf. «Differentsialnye uravneniya i smezhnye problemy» (Sterlitamak, 25–-29 iyunya 2018 g.), Ufa, 2018, 262–-264

[5] Virchenko Yu. P., Subbotin A. V., “Uravneniya dinamiki kondensirovannykh sred s lokalnym zakonom sokhraneniya”, Mat. V Mezhdunar. nauch. konf. «Nelokalnye kraevye zadachi i rodstvennye problemy matematicheskoi biologii, informatiki i fiziki» (Nalchik, 4–7 dekabrya 2018 g.), IPMA KBNTs RAN, Nalchik, 2018, 59

[6] Virchenko Yu. P., Subbotin A. V., “Opisanie klassa evolyutsionnykh uravnenii divergentnogo tipa dlya vektornogo polya”, Mat. IV Vseross. nauch.-prakt. konf. s mezhdunar. uchastiem «Sovremennye problemy fiziko-matematicheskikh nauk» (Orel, 22–25 noyabrya 2018 g.), Orel, 2018, 83–-86

[7] Virchenko Yu. P., Subbotin A. V., “Opisanie klassa evolyutsionnykh uravnenii ferrodinamiki”, Itogi nauki i tekhn. Ser. Sovr. mat. prilozh. Temat. obz., 170 (2019), 15–-30

[8] Gantmakher F. R., Teoriya matrits, Fizmatlit, M., 2004 | MR

[9] Godunov S. K., Uravneniya matematicheskoi fiziki, Nauka, M., 1979 | MR

[10] Gurevich G. B., Osnovy teorii algebraicheskikh invariantov, GITTL, M.-L., 1948 | MR

[11] Isaev A. A., Kovalevskii M. Yu., Peletminskii S. V., “O gamiltonovom podkhode k dinamike sploshnykh sred”, Teor. mat. fiz., 102:2 (1995), 283–-296 | MR | Zbl

[12] Isaev A. A., Kovalevskii M. Yu., Peletminskii S. V., “Gamiltonov podkhod v teorii kondensirovannykh sred so spontanno narushennoi simmetriei”, Fiz. elem. chast. atom. yadra., 27:2 (1996), 431–492

[13] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001

[14] Kulikovskii A. G., Slobodkina F. A., “Ravnovesie proizvolnykh statsionarnykh techenii v transzvukovykh tochkakh”, Prikl. mat. mekh., 31 (1968), 593–-602

[15] Kulikovskii A. G., Slobodkina F. A., “Ob ustoichivosti odnomernykh statsionarnykh reshenii giperbolicheskikh sistem differentsialnykh uravnenii pri nalichii tochek obrascheniya v nul odnoi iz kharakteristicheskikh skorostei”, Prikl. mat. mekh., 48:3 (1984), 414–-419 | MR

[16] Lyubarskii G. Ya., Teoriya grupp i ee prilozheniya v fizike, GIFML, M., 1958

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

[18] Mac-Connell A. J., Application of Tensor Analysis, Dover, New York, 1957 | MR

[19] Majda A., “The existence of multi-dimensional shock fronts”, Mem. Am. Math. Soc., 43:281 (1983), 1–-94 | MR

[20] Spencer A. G. M., “Theory of Invariants”, Continuum Physics, Academic Press, New York, 1971, 239–353 | MR