Geodesic
Hyperbolicity of covariant systems of first-order equations for vector and scalar fields
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 2, Tome 209 (2022), pp. 3-15.

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We consider a class of first-order systems of quasilinear partial differential equations $\dot{\boldsymbol{u}}=\mathsf{L}'[\boldsymbol{u},\boldsymbol{\rho}]$, $\dot{\boldsymbol{\rho}}=\mathsf {L}''[\boldsymbol{u},\boldsymbol{\rho}]$ that describe time evolution of the pair $\langle\boldsymbol{u},\boldsymbol{\rho}\rangle$ consisting of a vector field $\boldsymbol{u}(\boldsymbol{x},t)$ and the set of scalar fields $\boldsymbol{\rho}=\langle\rho^{(s)}(\boldsymbol{x},t);\ s=1,\dots,N\rangle$, $\boldsymbol{x}\in\mathbb{R}^3$. The class considered consists of systems that are invariant under time and space translations and covariant under space rotations. We describe the corresponding class of evolution generators, i.e., nonlinear first-order differential operators $\mathsf{L}=\langle\mathsf{L}'[\cdot],\mathsf{L}''[\cdot]\rangle$ acting in the functional space $C_{1,\mathrm{loc}}^{3+N}(\mathbb{R}^3)$. Also, we find conditions under which a pair of operators $\mathsf{L}$ generates a hyperbolic system.
Keywords: first-order differential operator, quasilinear system, hyperbolicity, vector field, covariance, spherical symmetry. 
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Yu. P. Virchenko; A. E. Novoseltseva. Hyperbolicity of covariant systems of first-order equations for vector and scalar fields. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 2, Tome 209 (2022), pp. 3-15. http://geodesic.mathdoc.fr/item/INTO_2022_209_a0/

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

[2] 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

[3] 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

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

[5] Virchenko Yu. P., Subbotin A. V., “Kovariantnye differentsialnye operatory pervogo poryadka”, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obz., 187 (2020), 19–30

[6] Virchenko Yu. P., Novoseltseva A. E., “Giperbolicheskie uravneniya pervogo poryadka v $\mathbb{R}^3$”, Mat. Mezhdunar. konf. «Sovremennye metody teorii funktsii i smezhnye problemy. Voronezhskaya zimnyaya matematicheskaya shkola» (Voronezh, 28 yanvarya – 2 fevralya 2021 g.), Voronezh, 2021, 81

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

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

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

[10] Spencer A. G. M., “Theory of invariants”, Continuum Physics, I. Part III, ed. Eringen A. C., Academic Press, New York, 1971, 239–353 | MR

[11] Virchenko Yu. P., Subbotin A. V., “The class of evolutionary ferrodynamic equations”, Math. Meth. Appl. Sci., 44 (2021), 11913–11922 | DOI | MR | Zbl