Asymptotic analysis of the Ginzburg-Landau model of superconductivity: reduction to a free boundary model

Author:
S. J. Chapman

Journal:
Quart. Appl. Math. **53** (1995), 601-627

MSC:
Primary 82D55; Secondary 35C20, 35Q55, 80A22

DOI:
https://doi.org/10.1090/qam/1359498

MathSciNet review:
MR1359498

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A detailed formal asymptotic analysis of the Ginzburg-Landau model of superconductivity is performed and it is found that the leading-order solution satisfies a vectorial version of the Stefan problem for the melting or solidification of a pure material. The first-order correction to this solution is found to contain terms analogous to those of surface tension and kinetic undercooling in the scalar Stefan model. However, the ``surface energy'' of a superconducting material is found to take both positive and negative values, defining type I and type II superconductors respectively, leading to the conclusion that the free boundary model is only appropriate for type I superconductors.

**[1]**J. Bardeen, L. N. Cooper, and J. R. Schreiffer, Phys. Rev.**108**, 1175 (1957)**[2]**G. Caginalp,*Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations*, Phys. Rev.**A39**, 5887 (1989)**[3]**G. Caginalp,*The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits*, IMA J. Appl. Math.**44**, 77-94 (1990)**[4]**S. J. Chapman,*Thesis*, Oxford University, 1991**[5]**S. J. Chapman,*Nucleation of superconductivity in decreasing fields*, I & II, Europ. J. Appl. Math.**5**, 449-494 (1994)**[6]**S. J. Chapman, S. D. Howison, J. B. McLeod, and J. R. Ockendon,*Normal-superconducting transitions in Ginzburg-Landau theory*, Proc. Roy. Soc. Edin.**119A**, 117-124 (1991)**[7]**S. J. Chapman, S. D. Howison, and J. R. Ockendon,*Macroscopic models of superconductivity*, SIAM Review**34**, No. 4, 529-560 (1992)**[8]**J. C. Crank,*Free and Moving Boundary Problems*, Oxford, 1984**[9]**A. B. Crowley and J. R. Ockendon,*Modelling mushy regions*, Appl. Sci. Res.**44**, 1-7 (1987)**[10]**Q. Du, M. D. Gunzburger, and S. Peterson,*Analysis and approximation of the Ginzburg-Landau model of superconductivity*, SIAM Review**34**, 54-81 (1992)**[11]**A. J. W. Duijvestijn,*On the transition from superconducting to normal phase, accounting for latent heat and eddy currents*, IBM J. Research & Development**3**, 2, 132-139 (1959)**[12]**T. E. Faber,*The intermediate state in superconducting plates*, Proc. Roy. Soc.**A248**, 461-481 (1958)**[13]**H. Frahm, S. Ullah, and A. T. Dorsey,*Flux dynamics and the growth of the superconducting phase*, Phys. Rev. Lett.**66**, 23, 3067-3070 (1991)**[14]**U. Essmann and H. Träuble,*The direct observation of individual flux lines in type*II*superconductors*, Phys. Lett.**A24**, 526 (1967)**[15]**V. L. Ginzburg and L. D. Landau,*On the theory of superconductivity*, Zh. Èksper. Teoret. Fiz.**20**, 1064 (1950)**[16]**L. P. Gor'kov and G. M. Éliashberg,*Generalisation of the Ginzburg-Landau equations for nonstationary problems in the case of alloys with paramagnetic impurities*, Soviet Phys. J.E.T.P.**27**, 328 (1968)**[17]**S. D. Howison, A. A. Lacey, and J. R. Ockendon,*Singularity development in moving boundary problems*, Quart. J. Mech. Appl. Math.**38**, 343-360 (1985)**[18]**J. B. Keller,*Propagation of a magnetic field into a superconductor*, Phys. Rev.**111**, 1497 (1958)**[19]**C. G. Kuper, Philos. Mag.**42**, 961 (1951)**[20]**F. Liu, M. Mondello, and N. Goldenfeld,*Kinetics of the superconducting transition*, Phys. Rev. Lett.**66**, 23, 3071-3074 (1991)**[21]**A. C. Rose-Innes and E. H. Rhoderick,*Introduction to superconductivity*, Pergamon, 1978**[22]**H. Träuble and U. Essmann,*Ein hochauflösendes Verfahren zur Untersuchung magnetischer Strukturen von Supraleitern*, Phys. Stat. Solids**18**, 813-828 (1966)**[23]**H. Träuble and U. Essmann,*Die Beobachtung magnetischer Strukturen von Supraleitern zweiter Art*, Phys. Stat. Solids**20**, 95-111 (1967)**[24]**M. Van Dyke,*Perturbation Methods in Fluid Mechanics*, The Parabolic Press, 1975

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
82D55,
35C20,
35Q55,
80A22

Retrieve articles in all journals with MSC: 82D55, 35C20, 35Q55, 80A22

Additional Information

DOI:
https://doi.org/10.1090/qam/1359498

Article copyright:
© Copyright 1995
American Mathematical Society