Expansion for the superheating field in a semi-infinite film in the weak-$\kappa$ limit

Castillo, Pierre Del

ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 6, pp. 971-993.

Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak-$\kappa$ limit. These authors deduced a formal expansion for the superheating field in powers of ${\kappa }^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr’s formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion in powers of ${\kappa }^{\frac{1}{2}}$ for the superheating field.

[1] C. Bolley and P. Del Castillo, Existence and uniqueness for the half-space Ginzburg-Landau model. Nonlinear Anal. 47/1 (2001) 135-146. | Zbl

[2] C. Bolley and B. Helffer, Rigorous results for the Ginzburg-Landau equations associated to a superconducting film in the weak $\kappa$-limit. Rev. Math. Phys. 8 (1996) 43-83. | Zbl

[3] C. Bolley and B. Helffer, Rigorous results on the Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field. Part II. Nonlinear Stud. 3 (1996) 1-32. | Zbl

[4] C. Bolley and B. Helffer, Proof of the De Gennes formula for the superheating field in the weak-$\kappa$ limit. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 597-613. | EuDML | Numdam | Zbl

[5] C. Bolley and B. Helffer, Superheating in a semi-infinite film in the weak-$\kappa$ limit: Numerical results and approximate models. ESAIM: M2AN 31 (1997) 121-165. | EuDML | Numdam | Zbl

[6] C. Bolley and B. Helffer, Upper bound for the solution of the Ginzburg-Landau equations in a semi-infinite superconducting field and applications to the superheating field in the large $\kappa$ regime. European J. Appl. Math. 8 (1997) 347-367. | Zbl

[7] C. Bolley, F. Foucher and B. Helffer, Superheating field for the Ginzburg-Landau equations in the case of a large bounded interval. J. Math. Phys. 41 (2000) 7263-7289. | Zbl

[8] S. Chapman, Superheating field of type II superconductors. SIAM J. Appl. Math. 55 (1995) 1233-1258. | Zbl

[9] P. Del Castillo, Thèse de doctorat. Université Paris-Sud (2000).

[10] P. Del Castillo, Two terms in the lower bound for the superheating field in a semi-infinite film in the weak-$\kappa$ limit. European J. Appl. Math. (2002). | MR | Zbl

[11] P.G. De Gennes, Superconductivity: Selected topics in solid state physics and theoretical Physics, in Proc. of 8th Latin american school of physics. Caracas (1966).

[12] V.L. Ginzburg, On the theory of superconductivity. Nuovo Cimento 2 (1955) 1234. | Zbl

[13] V.L. Ginzburg, On the destruction and the onset of superconductivity in a magnetic field. Zh. Èksper. Teoret. Fiz. 34 (1958) 113-125; Transl. Soviet Phys. JETP 7 (1958) 78-87. | Zbl

[14] Di Bartolo, T. Dorsey and J. Dolgert, Superheating fields of superconductors: Asymptotic analysis and numerical results. Phys. Rev. B 53 (1996); Erratum. Phys. Rev. B 56 (1997).

[15] W. Eckhaus, Matched asymptotic expansions and singular perturbations. North-Holland, Math. Studies 6 (1973). | MR | Zbl

[16] B. Helffer and F. Weissler, On a family of solutions of the second Painlevé equation related to superconductivity. European J. Appl. Math. 9 (1998) 223-243. | Zbl

[17] H. Parr, Superconductive superheating field for finite $\kappa$. Z. Phys. B 25 (1976) 359-361.

[18] M. Van Dyke, Perturbation Methods in fluid mechanics. Academic Press, Stanford CA (1975). | MR | Zbl

[19] B. Rothberg-Bibby, H.J. Fink and D.S. Mclachlan, First and second order phase transitions of moderately small superconductor in a magnetic field. North-Holland (1978).

[20] S. Kaplun, Fluid mechanics and singular perturbations. Academic Press (1967). | MR