Mass-Energy threshold dynamics for dipolar Quantum Gases
Abstract
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 blow-up 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.
Cite
@article{arxiv.2009.05933,
title = {Mass-Energy threshold dynamics for dipolar Quantum Gases},
author = {Van Duong Dinh and Luigi Forcella and Hichem Hajaiej},
journal= {arXiv preprint arXiv:2009.05933},
year = {2021}
}
Comments
33 pages. Communications in Mathematical Sciences, to appear