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Bose-Einstein condensation (BEC), first predicted by Einstein in 1925, has been realized experimentally in 1995 in alkali gases. The award of the 2001 Nobel Prize in Physics to E. Cornell, C. Wieman, and W. Ketterle acknowledged the importance of the achievement. In this new state of matter, which is very dilute and at very low temperature, a macroscopic fraction of the atoms occupy the same quantum level, and behave as a coherent matter wave similar to the coherent light wave produced by a laser. In the dilute limit, the condensate is well described by a mean-field theory and a macroscopic wave function. The properties of these gaseous quantum fluids have been the focus of international interest in physics, both experimentally and theoretically, and many applications are envisioned. An important issue is the relationship between BEC and superfluidity, in particular through the existence of vortices. The focus of this book is the mathematical properties of vortices, observed in very recent experiments on rotating condensates

This book contains results in three directions: the Thomas-Fermi regime or small ε problem, where there is a bounded number of vortices in the system (Chapters 3, 4, 6); the fast-rotation regime, which displays a vortex lattice (Chapter 5); and the experiment of a superfluid flow around an obstacle (Chapter 7). The tools and techniques needed to address these problems are very different: energy expansion using a small parameter for the Thomas-Fermi regime; double-scale convergence, homogenization techniques, and Fock-Bargmann space for the fast-rotating regime; energy estimates and nondegeneracy of a solution for the superfluid flow. The main mathematical results are summarized in the present chapter.

In this chapter, we want to study the shape of the minimizers =, of Where = (, ), = (−, ), (,∇)= (ū∇ - (∈ū)/2, ε is a small parameter, and Ω is the given rotational velocity. We assume that ρTF()= ρ0 −r is the disc of radius R= √ρ0 in (so that ρTF = 0 on ∂, and ∫ ρTF = 1, which prescribes the value of ρ0. The issue is to determine the number and location of vortices according to the value of Ω.

In this chapter, we are interested in the minimizers of the energy for varoius function ρTF(). As before, =(, ), ⊥ = (−, ),(, ∇) = (ū∇ - ∇ū)/2, ε is a small parameter, and Ω is the given rotational velocity. We assume that D = {ρTF > 0} and ρTF() describes respectively a nonradial harmonic confinement and a quartic trapping potential, that is, the model case are In case (4.3), for certain values of and , the domain becomes an annulus, and this changes the pattern of vortices.

When the velocity gets large, the size of the condensate and the number of vortices increase: a dense lattice is observed [1, 47, 58, 141], referred to as an Abrikosov lattice due to the analogy with superconductors. The description of the vortex lattice at high rotational velocity has been the focus of very recent papers of HO [79] and very recently by [64, 27, 49, 154, 147]. Our aim is to provide mathematical insight into the lattice pattern.

In this chapter, we are interested in a three-dimensional rotating condensate, in a setting similar to that of the experiments. In particular, we want to justify the observations of the bent vortices. Thus we want to study the shape of vortices in minimizers of the following energy: Where r=(, , ), Ω is parallel to the axis, ρ0 ⊂x+αy+βz. is the ellipsoid {ρTF > 0}={x+αty+βz < ρ0}, and ρ0 is determined by Which yields ρ0= 15αβ/8π. If β is small, as in the experiments, this gives rise to an elongated domain along the direction.

In this chapter, we address another issue related to superfluidity: the existence of a dissipationless flow induced by the motion of a macroscopic object in a superfluid. The nucleation of vortices corresponds to the breakdown of this dissipationless phenomenon.

Many open problems have been described in the course of the book, but we present some extra ones in this chapter, corresponding to new directions.