Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Uniqueness and Decay of Weak Solutions to Phase-Lock Equations

Discontinuity, Nonlinearity, and Complexity 10(1) (2021) 31--41 | DOI:10.5890/DNC.2021.03.003

Jishan Fan$^1$, Gen Nakamura$^2$, and Mei-Qin Zhan$^3$

$^1$ Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

$^2$ Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

$^3$ Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL32224, USA

Download Full Text PDF



In this paper, we prove the uniqueness of weak solutions $(f, Q)$ to the phase-lock equations with $f_0 \in L^2$ and $Q_0 \in L^3$ when the space dimension $d = 3.$ We also prove the uniqueness of weak solutions $(f, a)$ to the Ginzburg-Landau equations with $(f_0, a_0) \in L^p \times L^p$ and $1 < p < 2$ when $d = 1.$ We will also present a result on the decay of $Q$ as time $t\to\infty.$


  1. [1]  Zhan, M.Q. (2000), Phase-lock equations and its connections to Ginzburg-Landau equations of superconductivity, Nonlinear Analysis, 42, 1063-1075.
  2. [2]  Zhan, M.Q. (2005), Well-posedness of phase-lock equations of superconductivity, Appl. Math. Letters, 18(11), 1210-1215.
  3. [3]  Fan, J. and Ozawa, T. (2018), Global well-posedness of weak solutions to the time-dependent Ginzburg-Landau model for superconductivity, Taiwanese J. Math., 22(4) 851-858.
  4. [4]  Fan, J. and Ozawa, T. (2012), Uniqueness of weak solutions to the Ginzburg-Landau model for superconductivity, Zeit. Angew. Math. Phys., 63(3), 453-459.
  5. [5]  Fan, J., Gao, H., and Guo, B. (2015), Uniqueness of weak solutions to the 3D Ginzburg-Landau superconductivity model, Int. Math. Res. Notices, 2015(5), 1239-1246.
  6. [6]  Fan, J. and Gao, H. (2010), Uniqueness of weak solutions in critical space of the 3D time-dependent Ginzburg-Landau equations for supereconductivity, 283(8), 1134-1143.
  7. [7]  Xie, H.Y., Fan, J.S., Zhou, Y., and Sun, Y.S. (2020), Global well-posedness of weak and strong solutions to the $nD$ phase-lock system, Applicable Analysis, 1-6.
  8. [8]  Beir\~{a}o da Veiga, H. and Crispo, F. (2010), Sharp inviscid limit results under Navier type boundary conditions: An $L^p$ theory, Journal of Mathematical Fluid Mechanics, 12(3), 397-411.
  9. [9]  Beasley, M.R. (2009), Notes on the Ginzburg-Landau Theory, Notes on the Ginzburg-Landau Theory, ICMR Summer School on Novel Superconductors, University of California, Santa Barbara.
  10. [10] Konsin, P. and Sorkin, B. (2009), Time-dependent Ginzburg-Landau equations for a two-component superconductor and the doping dependence of the relaxation times of the order parameters in $YBa_2Cu_3O_{7-\delta}, $ Journal of Physics: Conference Series, 150, 052122.
  11. [11]  Adams, R.A. and Fournier, J.J.F. (2003), Sobolev Spaces, 2nd ed. Pure and Applied Mathematics (Amsterdam) 140. Amsterdam: Elsevier/Academic Press.
  12. [12]  Lunardi, A. (2009), Interpolation Theory, 2nd ed. Lecture Notes, Scuola Normale Superiore di Pisa (New Series). Pisa: Edizioni della Normale.
  13. [13]  Boccardo, L. and Gallou\"{e}t, T. (1989), Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87(1), 149-169.
  14. [14]  de Gennes, P. (1966), Superconductivity in Metals and Alloys, Benjamin, New York.