Global Existence and Uniqueness of Helically Symmetric Weak Solutions to the Ginzburg–Landau Model in Superconductivity

Fan, Jishan; Zhou, Yong

doi:10.3390/math14111873

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Global Existence and Uniqueness of Helically Symmetric Weak Solutions to the Ginzburg–Landau Model in Superconductivity

by

Jishan Fan

¹ and

Yong Zhou

^2,*

¹

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

²

Department of Mathematics, Wenzhou University, Wenzhou 350025, China

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(11), 1873; https://doi.org/10.3390/math14111873

Submission received: 27 March 2026 / Revised: 10 May 2026 / Accepted: 26 May 2026 / Published: 28 May 2026

(This article belongs to the Section C1: Difference and Differential Equations)

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Abstract

This manuscript proves the global existence and uniqueness of helically symmetric weak solutions to the 3D time-dependent Ginzburg–Landau model of superconductivity in

R^{3}

with

L^{2}

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 = n θ + α z

, 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

L^{2}

the new critical space.

Keywords: uniqueness; weak solutions; superconductivity

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MDPI and ACS Style

Fan, J.; Zhou, Y. Global Existence and Uniqueness of Helically Symmetric Weak Solutions to the Ginzburg–Landau Model in Superconductivity. Mathematics 2026, 14, 1873. https://doi.org/10.3390/math14111873

AMA Style

Fan J, Zhou Y. Global Existence and Uniqueness of Helically Symmetric Weak Solutions to the Ginzburg–Landau Model in Superconductivity. Mathematics. 2026; 14(11):1873. https://doi.org/10.3390/math14111873

Chicago/Turabian Style

Fan, Jishan, and Yong Zhou. 2026. "Global Existence and Uniqueness of Helically Symmetric Weak Solutions to the Ginzburg–Landau Model in Superconductivity" Mathematics 14, no. 11: 1873. https://doi.org/10.3390/math14111873

APA Style

Fan, J., & Zhou, Y. (2026). Global Existence and Uniqueness of Helically Symmetric Weak Solutions to the Ginzburg–Landau Model in Superconductivity. Mathematics, 14(11), 1873. https://doi.org/10.3390/math14111873

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Global Existence and Uniqueness of Helically Symmetric Weak Solutions to the Ginzburg–Landau Model in Superconductivity

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