@inbook{77be293ab4004325b4ceabb740bedb43,

title = "Dynamics of Solutions to the Gross–Pitaevskii Equation Describing Dipolar Bose–Einstein Condensates",

abstract = "We review some recent results on the long-time dynamics of solutions to the Gross–Pitaevskii equation (GPE) governing non-trapped dipolar quantum gases. We describe the asymptotic behaviors of solutions for different initial configurations of the initial datum in the energy space, specifically for data below, above, and at the mass–energy threshold. We revisit some properties of powers of the Riesz transforms by means of the decay properties of the integral kernel associated to the parabolic biharmonic equation. These decay properties play a fundamental role in establishing the dynamical features of the solutions to the studied GPE.",

author = "Jacopo Bellazzini and Luigi Forcella",

note = "Funding Information: Acknowledgments J.B. was partially supported by the project “Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari dispersive” of GNAMPA 2020, and by the project PRIN grant 2020XB3EFL. L.F. was supported by the EPSRC New Investigator Award (grant no. EP/S033157/1). Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.",

year = "2022",

month = aug,

day = "30",

doi = "10.1007/978-981-19-6434-3_2",

language = "English",

isbn = "9789811964336",

series = "Springer INdAM Series",

publisher = "Springer",

pages = "25--57",

booktitle = "Qualitative Properties of Dispersive PDEs. INdAM 2021",

}