Catherine Bolley

RIGOROUS RESULTS FOR THE GINZBURG-LANDAU EQUATIONS ASSOCIATED TO A SUPERCONDUCTING FILM IN THE WEAK κ LIMIT

In continuation with our preceding paper [10] concerning the superconducting film, we present in this article rigorous results concerning the superheating in the weak κ limit. The principal result is an important step toward the rigorous proof of a formula due to P. De Gennes [26] . This paper is complementary to our paper [11] where numerical results are presented and approximate models are discussed. Most of the results have been announced in [12] and [13].

Proof of the De Gennes formula for the superheating field in the weak κ limit

Abstract In continuation with our preceding paper [3] concerning the superconducting film, we present in this article new estimates for the superheating field in the weak κ limit. The principal result is the proof of the existence of a finite superheating field h sh ,+ ( κ ) (obtained by restricting the usual definition of the superheating field to solutions of the Ginzburg-Landau system ( f , A ) with f positive) in the case of a semi-infinite interval. The bound is optimal in the limit κ → 0 and permits to prove (combining with our previous results) the De Gennes formula Δu−β(x 1 ,y,c)∂ 1 u + ƒ(x 1 ,u) = 0 in σ ∂ ν u = 0 on ∂σ u(−∞·) = 0, u(+∞,·) = 1 The proof is obtained by improving slightly the estimates given in [3] where an upper bound was found but under the additional condition that the function f was bounded from below by some fixed constant ρ > 0.

Stability of Bifurcating Solutions for the Ginzburg–Landau Equations

This paper is concerned with superconducting solutions of the Ginzburg–Landau equations for a film. We study the structure and the stability of the bifurcating solutions starting from normal solutions as functions of the parameters (κ, d), where d is the thickness of the film and κ is the Ginzburg–Landau parameter characterizing the material. Although κ and d play independent roles in the determination of these properties, we will exhibit the dominant role taken up by the product κd in the existence and uniqueness of bifurcating solutions as much as in their stability. Using the semi-classical analysis developed in our previous papers for getting the existence of asymmetric solutions and asymptotics for the supercooling field, we prove in particular that the symmetric bifurcating solutions are stable for (κ, d) such that κd is small and (for any η>0) and unstable for (κ, d) such that κd is large. We also show the existence of an explicit critical value Σ0 such that, for κ≤Σ0-η and κd large, the asymmetric solutions are unstable, while, for κ≥Σ0+η and κd large, the asymmetric solutions are stable. Finally, we also analyze the symmetric problem which leads to other stability results.

Superheating field for the Ginzburg–Landau equations in the case of a large bounded interval

We study the asymptotic behavior of the local superheating field for a film of width 2d in the regime κ small, κd large, where κ is the Ginzburg–Landau parameter. This gives a mathematical justification for the introduction of the semi-infinite model as a good approximation for this regime.

A Priori Estimates for Ginzburg-Landau Solutions

We analyze some aspects of the structure of the set (f(0), A’(0)) where (f, A) is a solution of the Ginzburg-Landau equation for the semi-infinite model. We analyze also how this semi-infinite model gives a good information on the behavior of a large film. We emphasize about the techniques of a priori estimates developed in [11], [12], [13], [6] and present also new estimates relating f(0) and A’(0).

ON THE CONTINUITY OF THE RESPONSE MAP AND AN ALTERNATIVE SHOOTING METHOD FOR THE HALF-SPACE GINZBURG–LANDAU MODEL

As a consequence of a rather complete analysis of the qualitative properties of the solutions of the Ginzburg–Landau equations, we prove, in this paper, both the continuity of a fundamental map σ, called response map in the physical literature on superconductors, and the convergence of an efficient algorithm for the computation of the graph of σ. The response map σ gives the intensity h of the external magnetic field for which the Ginzburg–Landau equations (in a half-space) have a solution such that the parameter order has a prescribed value at the boundary of the sample. Our study involves a shooting method on either one or the other unknown of the system; our algorithm has been introduced in Bolley–Helffer for small values of the Ginzburg–Landau parameter κ and extended in Bolley to any value of κ. Our preceding mathematical studies were not sufficient to prove the convergence, but a recent result (in Ref. 3) on the monotonicity of the solutions with respect to h, combined with a more extensive use of the properties of the solutions of the Ginzburg–Landau system, allow us to complete the proof and to get, as a by-product, the continuity of σ.

Global superheating field for superconductors in a large bounded interval

Abstract In continuation of our study of the Ginzburg–Landau equations, we study the asymptotic behavior of the superheating field for a film of width 2d in the regime κ small, κd large, where κ is the Ginzburg–Landau parameter. Using new estimates on the inner magnetic potential, we improve our conditions on the parameter κd and we prove that the local superheating field analyzed in our preceding paper [C. Bolley, F. Foucher, B. Helffer, Superheating field for the Ginzburg–Landau equations in the case of a large bounded interval, J. Math. Phys. 41 (11) (2000) 7263–7289] is a global one. In the regime κ and d large, we also give accurate estimates on the superheating field.

ASYMPTOTICS FOR CRITICAL FIELDS IN AN APPROXIMATE GINZBURG–LANDAU MODEL IN THE LARGE SIZE LIMIT

Pursuing a study of the superheating field for the Ginburg–Landau equations performed in Ref. 8, we consider an approximate model for a film on a bounded interval when the thickness 2e is large in comparison with the inverse of the characteristic κ of the material. We show for this problem the existence of a local superheating field, and recover, as e tends to +∞, the superheating field for a similar approximate model on a semi-infinite interval which has been studied in Ref. 8.