×

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.

MSC:

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
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.