Vortices are often associated with dramatic circumstances such as hurricanes; this type of vortex has been studied extensively in the framework of classical fluids. Nev- ertheless, its quantum counterpart has gained major interest in the past few years due to the experimental realization of Bose–Einstein condensates (BEC), a new state of matter predicted by Einstein in 1925. Vortices in BEC are quantized, and their size, origin, and significance are quite different from those in normal fluids since they exemplify “superfluid” properties.
Since the first experimental achievement of Bose–Einstein condensates in 1995 in alkali gases and the award of the Nobel Prize in Physics in 2001, the properties of these gaseous quantum fluids have been the focus of international interest in con- densed matter physics. This book was both motivated by this intense activity, espe- cially in the group of Jean Dalibard at the Ecole normale supe ́rieure, but also by the constant development of mathematical techniques which could prove useful in tack- ling these problems, in particular in the group of Haim Brezis. This monograph is dedicated to the mathematical modelling of some specific experiments which display vortices and to a rigorous analysis of features emerging experimentally. It can serve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum condensates, and can also complement a graduate semi- nar in elliptic PDEs or modelling of physical experiments. There are two introductory chapters: the first is related to the physics background, while the second is devoted to the presentation of the mathematical results described in the book.