This manuscript proves the global existence and uniqueness of helically symmetric weak solutions to the 3D time-dependent Ginzburg–Landau model of superconductivity in R3 with L2 initial data and supports the choice of the Lorentz gauge. For this, we define helical symmetry using cylindrical coordinates (r,θ,z) and a helical variable ξ=nθ+αz, where n is an even integer and α>0. This reduction allows the 3D system to be treated with specific geometric constraints. The imposition of helical symmetry effectively reduces the dimension of the problem, thereby making L2 the new critical space.
Fan et al. (Thu,) studied this question.