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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2010, Volume 16, Number 4, Pages 128–143
(Mi timm648)
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This article is cited in 2 scientific papers (total in 3 papers)
The class of solenoidal planar-helical vector fields
V. P. Vereshchagina, Yu. N. Subbotinb, N. I. Chernykhb a Russian State Professional Pedagogical University
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Abstract:
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, Gromeka's problem.
Received: 22.01.2010
Citation:
V. P. Vereshchagin, Yu. N. Subbotin, N. I. Chernykh, “The class of solenoidal planar-helical vector fields”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 4, 2010, 128–143; Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S171–S187
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https://www.mathnet.ru/eng/timm648 https://www.mathnet.ru/eng/timm/v16/i4/p128
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