Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 1

Global attractor for a Ginzburg–Landau type model of rotating Bose–Einstein condensates

Pages: 5 – 32

DOI: http://dx.doi.org/10.4310/DPDE.2017.v14.n1.a2

Authors

Alexey Cheskidov (Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.)

Daniel Marahrens (Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany)

Christof Sparber (Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.)

Abstract

We study the long time behavior of solutions to a nonlinear partial differential equation arising in the mean-field description of trapped rotating Bose–Einstein condensates. The equation can be seen as a hybrid between the well-known nonlinear Schrödinger/Gross–Pitaevskii equation and the Ginzburg–Landau equation. We prove existence and uniqueness of global in-time solutions in the physical energy space and establish the existence of a global attractor within the associated dynamics. We also obtain basic structural properties of the attractor and an estimate on its Hausdorff and fractal dimensions. As a by-product, we establish heat-kernel estimates on the linear part of the equation.

Keywords

Gross–Pitaevskii equation, Bose–Einstein condensation, Ginzburg–Landau equation, vortices, global attractor

2010 Mathematics Subject Classification

35A01, 35Q55

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Published 31 January 2017