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# 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

#### 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

Published 31 January 2017