The class of solenoidal planar-helical vector fields

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 planarhelical solutions of Gromeka’s problem in some domain D ⊂ 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 specific solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonic conjugates in D ² = D∩R ²; i.e., Gromeka’s nonlinear problem is reduced to linear boundary value problems. As an example, a specific solution of the problem for an axisymmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in $\bar D^2$ in terms of wavelet systems that form bases of various spaces of functions harmonic in D ².