Catálogo de publicaciones - libros

This book is devoted to the mathematical study of the two-dimensional Ginzburg-Landau model with magnetic field. This is a model of great importance and recognition in physics (with several Nobel prizes awarded for it: Landau, Ginzburg, and Abrikosov). It was introduced by Ginzburg and Landau (see [101]) in the 1950s as a phenomenological model to describe superconductivity. Superconductivity was itself discovered in 1911 by Kammerling Ohnes. It consists in the complete loss of resistivity of certain metals and alloys at very low temperatures. The two most striking consequences of it are the possibility of permanent and the particular behavior that, when the material is submitted to an external magnetic field, that field gets expelled from it. Aside from explaining these phenomena, and through the very influential work of A. Abrikosov [1], the Ginzburg-Landau model allows one to predict the possibility of a in type II superconductors where triangular vortex lattices appear. These vortices—in a few words a vortex can be described as a quantized amount of vorticity of the superconducting current localized near a point—have since been the objects of many observations and experiments. The first observation dates back from 1967, by Essman and Trauble, see [93]. For pictures of lattice observations in superconductors and more references to experimental results, refer to the web page .

We begin by describing how the expression (1.1) for the Ginzburg-Landau functional is deduced from the expression (2.1) below, more commonly found in the physics literature. We will also give a nonrigorous introduction to critical fields in ℝ, in the spirit of Abrikosov, and draw a corresponding phase diagram in the () plane, i.e., qualitatively describe minimizers of the Ginzburg-Landau energy for different values of and , emphasizing the role of the vortices. Three areas of the parameter plane will be found: the normal, superconducting and mixed states, separated by what are usually called .

In this chapter, we start to investigate the mathematical aspects of the Ginzburg-Landau energy and equations. Whereas the material in the first three sections is relatively easy or standard (existence of minimizers, regularity of solutions, apriori estimates …) and used throughout the later chapters, the material of the last two sections is more advanced, contains several results stated without proofs, and is only used in Chapter 5 and then Chapters 10 to 12. However, we feel that the material is important enough, like the uniqueness result of P. Mironescu (Theorem 3.2), or basic enough to deserve to be stated early on.

The aim of this chapter is to provide one of the basic tools for the analysis of the Ginzburg-Landau functional in terms of vortices.

The key ingredient here is the Pohozaev identity for solutions of Ginzburg- Landau. This identity was already used crucially in Bethuel-Brezis- Hélein [43], Brezis-Merle-Rivière [61], and its first use on small balls goes back to Bethuel-Rivière [52] and Struwe [189]. Its consequences were also explored further in the book of Pacard-Rivière [148]. Here, the idea is to combine it with the ball-construction method in order to obtain lower bounds for the energy in terms of the potential term ∫(1 − ||) instead of the degree, or equivalently, upper bounds of the potential by the energy divided by | log |. This method works for solutions of the Ginzburg-Landau equation, without magnetic field as well as with. We will present the two situations in parallel, in Sections 5.1 and 5.2. In the third section of the chapter, we present applications to the microscopic analysis of vortices of solutions of () or (1.3). Among all these results, only Theorem 5.4 will be used later, for the study of solutions with bounded numbers of vortices: for Proposition 10.2 and in the course of the proof of Theorem 11.1.

In this chapter we show that the vortex balls provided by Theorem 4.1, although they are constructed through a complicated process and are not completely intrinsic to () (and not unique), have in the end a simple relation to the configuration (), namely that the measure Σ2δ is close in a certain norm to the gauge-invariant version of the Jacobian determinant of , an intrinsic quantity depending on (). This will allow us, in the next chapters, to extract from (), in addition to the vortex energy πΣ |||log | contained in the vortex balls, a term describing vortex-vortex interactions and vortex-applied field interactions in terms of the measure Σ2πdδ.

In this chapter, we start studying the question of minimizing the energy and we prove the main result of Γ-convergence of . As already mentioned, configurations have a vorticity (,), which, according to Chapter 6, is compact as → 0 (under a suitable energy bound) and the result we obtain below shows that minimizers of have vorticities which converge to a measure which minimizes a certain convex energy. This measure, by convex duality, is shown to be the solution to a simple .

The previous chapter dealt with minimizers of the Ginzburg-Landau functional when the applied field was (|log |). The applied field behaving asymptotically like λ|log |, letting λ → ∞ in Theorem 7.2 indicates that for energy-minimizers for applied fields ≫ |log |, we must have . But in this regime, and the arguments of Chapter 7 do not give, even formally, the leading order term of the minimal energy. Moreover, the tools which were at the heart of the result, namely the vortex balls construction of Theorem 4.1 and the Jacobian estimate of Theorem 6.1 break down for higher values of .

When ∼ i.e., , then from Theorem 7.2, we get that the limiting minimizer is hence μ* = 0. Moreover, comparing the lower bounds (7.58) and (7.59) to the upper bound of Theorem 7.1, we find , which means that the number of vortices is (). In other words, for energy-minimizers, vortices first appear for , or ∼ , and next to Λ (defined in (7.5)), and the vorticity mass is much smaller than . The analysis of Chapter 7 does not give us the optimal number of vortices nor the full asymptotic expansion of the first critical field. Thus, a more detailed study will be necessary in this regime , in which ≪ . We will prove that the vortices, even though their number may be diverging, all concentrate around Λ (generically a single point) but that after a suitable blow-up, they tend to arrange in a uniform density on a subdomain of ℝ, in order to minimize a limiting interaction energy defined on probability measures.

In this chapter, we prove upper bound and lower bound estimates for configurations with a number of vortices as → 0 which reduces to considering a number of vortices . These estimates will be useful in the next chapter. The fact that the number of vortices is bounded independently of allows us to obtain much more precise information with specific techniques: the upper and lower bounds will match up to an error which is (1) as → 0.