Geodesic
A solution class of the Euler equation in a~torus with solenoidal velocity field
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 4, pp. 60-70

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A system of equations with respect to a pair $(\mathbf V,p)$ of a scalar field and a vector field in a torus $D$ is considered. The system consists of the Euler equation with a given vector field $\mathbf f$ and the solenoidality equation for the field $\mathbf V$. We seek for solutions $(\mathbf V,p)$ of this system for which lines of the vector field $\mathbf V$ inside $D$ coincide with meridians of tori embedded in $D$ with the same circular axis. Conditions on the vector field $\mathbf f$ under which the problem is solvable are established, and the whole class of such solutions is described.
Keywords: scalar and vector fields, curl.
Mots-clés : Euler equation, divergence 
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     author = {V. P. Vereshchagin and Yu. N. Subbotin and N. I. Chernykh},
     title = {A solution class of the {Euler} equation in a~torus with solenoidal velocity field},
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     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/TIMM_2014_20_4_a5/}
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V. P. Vereshchagin; Yu. N. Subbotin; N. I. Chernykh. A solution class of the Euler equation in a~torus with solenoidal velocity field. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 4, pp. 60-70. http://geodesic.mathdoc.fr/item/TIMM_2014_20_4_a5/