Geodesic
The class of solenoidal planar-helical vector fields
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 128-143
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The class of solenoidal vector fields whose lines lie in planes parallel to $R^2$ is constructed by the method of mappings. This class exhausts the set of all smooth planar-helical solutions of Gromeka's problem in some domain $D\subset R^3$. In the case of domains $D$ with cylindrical boundaries whose generators are orthogonal to $R^2$, it is shown that the choice of a concrete solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonically conjugate in $D^2=D\cap R^2$; i.e., Gromeka's nonlinear problem is reduced to linear boundary value problems. As an example, a concrete solution of the problem for an axially symmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in $\overline D^2$ in terms of wavelet systems that form bases of various spaces of functions harmonic in $D^2$.
Keywords: scalar fields, vector fields, tensor fields, curl, wavelets
Mots-clés : Gromeka's problem. 
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V. P. Vereshchagin; Yu. N. Subbotin; N. I. Chernykh. The class of solenoidal planar-helical vector fields. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 128-143. http://geodesic.mathdoc.fr/item/TIMM_2010_16_4_a12/

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