Geodesic
Second-order evolution equations of divergent type for solenoidal vector fields on $\mathbb{R}^3$
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 198 (2021), pp. 41-49

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In this paper, we describe the class $\mathfrak{K}_2^{(0)}(\mathbb{R}^3)$ of second-order differential operators of divergent type that are invariant under translations of $\mathbb{R}^3$ and are transformed covariantly under rotations of $\mathbb{R}^3$. Using such operators, one can construct evolutional equations that describe a translation-invariant dynamics of a solenoidal vector field $\boldsymbol{V}(\boldsymbol{x},t)$ so that each operator of the class $\mathfrak{K}_2^{(0)}(\mathbb{R}^3)$ determines an infinitesimal $t$-shift of this field. Also, we prove that the class of all evolutional equations for a unimodal vector field $\boldsymbol{V}(\boldsymbol{x},t)$ is trivial.
Keywords: divergent differential operator, translational invariance, vector field, covariance, field flux density, unimodality, solenoidality. 
@article{INTO_2021_198_a3,
     author = {Yu. P. Virchenko and A. V. Subbotin},
     title = {Second-order evolution equations of divergent type for solenoidal 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},
     pages = {41--49},
     publisher = {mathdoc},
     volume = {198},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_198_a3/}
}
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Yu. P. Virchenko; A. V. Subbotin. Second-order evolution equations of divergent type for solenoidal vector fields on $\mathbb{R}^3$. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 198 (2021), pp. 41-49. http://geodesic.mathdoc.fr/item/INTO_2021_198_a3/