Global well-posedness of weak solutions to the time-dependent Ginzburg-Landau model for superconductivity. (English) Zbl 1404.35425

Summary: We prove the global existence and uniqueness of weak solutions to the time dependent Ginzburg-Landau system in superconductivity with Coulomb gauge.


35Q56 Ginzburg-Landau equations
35K55 Nonlinear parabolic equations
82D55 Statistical mechanics of superconductors
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second ed, Pure and Applied Mathematics (Amsterdam) 140, Elsevier/Academic Press, Amsterdam, 2003.
[2] H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions: An \(L^p\) theory, J. Math. Fluid Mech. 12 (2010), no. 3, 397–411. · Zbl 1261.35099
[3] A. Bendali, J. M. Domínguez and S. Gallic, A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains, J. Math. Anal. Appl. 107 (1985), no. 2, 537–560. · Zbl 0591.35053
[4] Z. Chen, C. M. Elliott and T. Qi, Justification of a two-dimensional evolutionary Ginzburg-Landau superconductivity model, RAIRO Modél. Math. Anal. Numér. 32 (1998), no. 1, 25–50. · Zbl 0905.35084
[5] Z. M. Chen, K.-H. Hoffmann and J. Liang, On a nonstationary Ginzburg-Landau superconductivity model, Math. Methods Appl. Sci. 16 (1993), no. 12, 855–875. · Zbl 0817.35111
[6] Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal. 53 (1994), no. 1-2, 1–17. · Zbl 0843.35019
[7] J. Fan, H. Gao and B. Guo, Uniqueness of weak solutions to the 3D Ginzburg-Landau superconductivity model, Int. Math. Res. Not. IMRN 2015 (2015), no. 5, 1239–1246. · Zbl 1317.35248
[8] J. Fan and S. Jiang, Global existence of weak solutions of a time-dependent 3-D Ginzburg-Landau model for superconductivity, Appl. Math. Lett. 16 (2003), no. 3, 435–440. · Zbl 1055.35109
[9] J. Fan and T. Ozawa, Global strong solutions of the time-dependent Ginzburg-Landau model for superconductivity with a new gauge, Int. J. Math. Anal. (Ruse) 6 (2012), no. 33-36, 1679–1684. · Zbl 1255.35202
[10] ——–, Uniqueness of weak solutions to the Ginzburg-Landau model for superconductivity, Z. Angew. Math. Phys. 63 (2012), no. 3, 453–459. · Zbl 1247.35164
[11] A. Lunardi, Interpolation Theory, Second edition, Lecture Notes, Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009.
[12] Q. Tang, On an evolutionary system of Ginzburg-Landau equations with fixed total magnetic flux, Comm. Partial Differential Equations 20 (1995), no. 1-2, 1–36. · Zbl 0833.35132
[13] Q. Tang and S. Wang, Time dependent Ginzburg-Landau equations of superconductivity, Phys. D 88 (1995), no. 3-4, 139–166. · Zbl 0900.35371
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