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Asymptotic analysis of the Ginzburg-Landau model of superconductivity: Reduction to a free boundary model. (English) Zbl 0842.35120

Summary: 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.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
82D55 Statistical mechanics of superconductors
35R35 Free boundary problems for PDEs
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