Communications in Mathematical Sciences
Volume 20 (2022)
Mass-energy threshold dynamics for dipolar quantum gases
Pages: 165 – 200
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