Communications in Mathematical Sciences

Volume 20 (2022)

Number 1

Mass-energy threshold dynamics for dipolar quantum gases

Pages: 165 – 200



Van Duong Dinh (Laboratoire Paul Painlevé, Université de Lille, Villeneuve d’Ascq, France; and Department of Mathematics, HCMC University of Education, Ho Chi Minh, Vietnam)

Luigi Forcella (Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Switzerland)

Hichem Hajaiej (Department of Mathematics, California State University, Los Angeles, Calif., U.S.A.)


We consider a Gross–Pitaevskii equation which appears as a model in the description of dipolar Bose–Einstein condensates, without a confining external trapping potential. We describe the asymptotic dynamics of solutions to the corresponding Cauchy problem in the energy space in different configurations with respect to the mass-energy threshold, namely for initial data above and at the mass-energy threshold. We first establish a scattering criterion for the equation that we prove by means of the concentration/compactness and rigidity scheme. This criterion enables us to show the energy scattering for solutions with data above the mass-energy threshold, for which only blowup was known. We also prove a blow-up/grow-up criterion for the equation with general data in the energy space. As a byproduct of scattering and blow-up criteria, and the compactness of minimizing sequences for the Gagliardo–Nirenberg’s inequality, we study long-time dynamics of solutions with data lying exactly at the mass-energy threshold.


Gross–Pitaevskii equation, dipolar BEC, energy scattering, finite-time blow-up, concentration phenomena

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35B40, 35B44, 82C10.

Received 17 September 2020

Received revised 8 June 2021

Accepted 8 June 2021

Published 10 December 2021