Skip to main content

A mean-field model of superconducting vortices

  • S. J. Chapman (a1), J. Rubinstein (a2) and M. Schatzman (a3)

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.

Hide All
[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).
[2] Chapman, S. J., Howison, S. D. & Ockendon, J. R. 1992 Macroscopic models of superconductivity. SIAM Review 34 (4), 529560.
[3] Keller, J. B. 1958 Propagation of a magnetic field into a superconductor. Phys. Rev. 111, 14971499.
[4] Abrikosov, A. A. 1957 On the magnetic properties of superconductors of the second group. Soviet Phys. J.E.T.P. 5 (6), 11741182.
[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.
[6] Chapman, S.J. 1994 Nucleation of superconductivity in decreasing fields I. Euro. J. Appl. Math. 5, 449468.
[7] Chapman, S.J. 1994 Nucleation of superconductivity in decreasing fields II. Euro. J. Appl. Math. 5, 469494.
[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), 12261227.
[9] Millman, M. H. & Keller, J. B. 1969 Perturbation theory of nonlinear boundary-value problems. J. Math. Phys. 10 (2), 342.
[10] Odeh, F. 1967 Existence and bifurcation theorems for the Ginzburg–Landau equations. J. Math. Phys. 8(12), 23512356.
[11] Chapman, S. J. 1995 Superheating field of type-11 superconductors. SIAM J. Appl. Math. 55 (5), 12331258.
[12] Ginzburg, V. L. & Landau, L. D. 1950 On the theory of superconductivity. Soviet Phys. J.E.T.P. 20, 1064.
[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, 259295.
[14] Peres, L. & Rubinstein, J. 1993 Vortex dynamics in U(1) Ginzburg–Landau models. Physica D 64 (1–3): 299309.
[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, 83768386.
[16] Rubinstein, J. & Keller, J. B. 1989 Particle distribution functions in suspensions. Phys. Fluids Al, 16321641.
[17] Carlson, N.-N. 1991 A topological defect model of superfluid vortex filaments. PhD Thesis, University of California, Berkeley.
[18] Crandall, M. G. & Rabinovich, P. H. 1973 Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rat. Mech. Anal. 52, 161180.
[19] Gilbarg, D. N., & Trudinger, N. S. 1977 Elliptic Partial Differential Equations of Second Order. Springer-Verlag.
[20] Kinderlehrer, D. & Nirenberg, L. 1977 Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa 4, 373391.
[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), 328334.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed