Geodesic
Hyperbolic quasilinear covariant first-order equations of divergent type for vector fields on $\mathbb{R}^3$
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 2, Tome 191 (2021), pp. 16-28.

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In this paper, we present a complete description of the class of first-order hyperbolic quasilinear equations of divergent type that describe the change in time $t\in\mathbb{R}$ of vector fields $\boldsymbol{v}(\boldsymbol{x},t)$, $\boldsymbol{x}\in\mathbb{R}^3$, which are invariant under translations in time $t\in\mathbb{R}$ and space $\mathbb{R}^3$ and transform covariantly under transformations from the group $\mathbb{O}_3$ of rotations of the space $\mathbb{R}^3$. This class is compared with the class of similar equations, which are hyperbolic in the sense of Friedrichs.
Keywords: quasilinear system, hyperbolicity, translational invariance, vector field, covariance, flux density.
Mots-clés : equation of divergent type 
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     title = {Hyperbolic quasilinear covariant first-order equations of divergent type for vector fields on $\mathbb{R}^3$},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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Yu. P. Virchenko; A. V. Subbotin. Hyperbolic quasilinear covariant first-order equations of divergent type for vector fields on $\mathbb{R}^3$. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 2, Tome 191 (2021), pp. 16-28. http://geodesic.mathdoc.fr/item/INTO_2021_191_a2/

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