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A mean-field model of superconducting vortices

Published online by Cambridge University Press:  26 September 2008

S. J. Chapman ,
J. Rubinstein  and
M. Schatzman
S. J. Chapman
Affiliation:
Mathematical Institute, 24–29 St. Giles', Oxford 0X1 3LB, UK
J. Rubinstein
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
M. Schatzman
Affiliation:
Analyse Numérique, U.R.A. 740 du C.N.R.S., Université Lyon l Claude-Bernard, 69622 Villeurbanne, France
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Abstract

A mean-field model for the motion of rectilinear vortices in the mixed state of a type-II superconductor is formulated. Steady-state solutions for some simple geometries are examined, and a local existence result is proved for an arbitrary smooth geometry. Finally, a variational formulation of the steady-state problem is given which shows the solution to be unique.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 2 , April 1996 , pp. 97 - 111
Copyright
Copyright © Cambridge University Press 1996

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References

[1] Chapman, S.J. 1995 Asymptotic analysis of the Ginzburg–Landau model of superconductivity: Reduction to a free boundary model. Quart. Appl. Math. (to appear).CrossRefGoogle Scholar
[2] Chapman, S. J., Howison, S. D. & Ockendon, J. R. 1992 Macroscopic models of superconductivity. SIAM Review 34 (4), 529–560.CrossRefGoogle Scholar
[3] Keller, J. B. 1958 Propagation of a magnetic field into a superconductor. Phys. Rev. 111, 1497–1499.CrossRefGoogle Scholar
[4] Abrikosov, A. A. 1957 On the magnetic properties of superconductors of the second group. Soviet Phys. J.E.T.P. 5 (6), 1174–1182.Google Scholar
[5] Bolley, C. & Helffer, B. 1994 Rigorous results on Ginzburg–Landau models in a film submitted to an exterior parallel magnetic field. Preprint Ecole centrale de Nantes.Google Scholar
[6] Chapman, S.J. 1994 Nucleation of superconductivity in decreasing fields I. Euro. J. Appl. Math. 5, 449–468.CrossRefGoogle Scholar
[7] Chapman, S.J. 1994 Nucleation of superconductivity in decreasing fields II. Euro. J. Appl. Math. 5, 469–494.CrossRefGoogle Scholar
[8] Kleiner, W. H., Roth, L. M. & Autler, S. H. 1964 Bulk solution of Ginzburg–Landau equations for type-II superconductors: Upper critical field region. Phys. Rev. 133 (5A), 1226–1227.CrossRefGoogle Scholar
[9] Millman, M. H. & Keller, J. B. 1969 Perturbation theory of nonlinear boundary-value problems. J. Math. Phys. 10 (2), 342.CrossRefGoogle Scholar
[10] Odeh, F. 1967 Existence and bifurcation theorems for the Ginzburg–Landau equations. J. Math. Phys. 8(12), 2351–2356.CrossRefGoogle Scholar
[11] Chapman, S. J. 1995 Superheating field of type-11 superconductors. SIAM J. Appl. Math. 55 (5), 1233–1258.CrossRefGoogle Scholar
[12] Ginzburg, V. L. & Landau, L. D. 1950 On the theory of superconductivity. Soviet Phys. J.E.T.P. 20, 1064.Google Scholar
[13] Berger, M. S. & Chen, Y. Y. 1989 Symmetric vortices for the Ginzburg–Landau equations of superconductivity and the nonlinear desingularization phenomenon. J. Fund. Anal. 82, 259–295.CrossRefGoogle Scholar
[14] Peres, L. & Rubinstein, J. 1993 Vortex dynamics in U(1) Ginzburg–Landau models. Physica D 64 (1–3): 299–309.CrossRefGoogle Scholar
[15] Dorsey, A. 1992 Vortex motion and the Hall effect in type-II superconductors: a time dependent Ginzburg–Landau theory approach. Phys. Rev. B 46, 8376–8386.CrossRefGoogle ScholarPubMed
[16] Rubinstein, J. & Keller, J. B. 1989 Particle distribution functions in suspensions. Phys. Fluids Al, 1632–1641.CrossRefGoogle Scholar
[17] Carlson, N.-N. 1991 A topological defect model of superfluid vortex filaments. PhD Thesis, University of California, Berkeley.Google Scholar
[18] Crandall, M. G. & Rabinovich, P. H. 1973 Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rat. Mech. Anal. 52, 161–180.CrossRefGoogle Scholar
[19] Gilbarg, D. N., & Trudinger, N. S. 1977 Elliptic Partial Differential Equations of Second Order. Springer-Verlag.CrossRefGoogle Scholar
[20] Kinderlehrer, D. & Nirenberg, L. 1977 Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa 4, 373–391.Google Scholar
[21] Gor'kov, L. P. & EĹiashberg, G. M. 1968 Generalization of the Ginzburg–Landau equations for non-stationary problems in the case of allows with paramagnetic impurities. Soviet Phys. J.E.T.P. 27(2), 328–334.Google Scholar