Quantum hydrodynamics with nonlinear interactions

Paolo  Antonelli; Pierangelo  Marcati

doi:10.3934/dcdss.2016.9.1

Article Contents

2016, Volume 9, Issue 1: 1-13. Doi: 10.3934/dcdss.2016.9.1

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Quantum hydrodynamics with nonlinear interactions

Paolo Antonelli^1, and
Pierangelo Marcati^2,

1.
Gran Sasso Science Institute, viale F. Crispi, 7, 67100 L'Aquila
2.
DISIM - Università de L'Aquila, via Vetoio, 67010 Coppito (AQ)

Received: August 31, 2014

Revised: January 31, 2015

Published: January 2016

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Abstract

In this paper we prove the global existence of large amplitude finite energy solutions for a system describing Quantum Fluids with nonlinear nonlocal interaction terms. The system may also (but not necessarily) include dissipation terms which do not provide any help to get the global existence. The method is based on the polar factorization of the wave function (which somehow generalizes the WKB method), the construction of approximate solutions via a fractional step argument and the deduction of Strichartz type estimates for the approximate solutions. Finally local smoothing and a compactness argument of Lions Aubin type allow to show the convergence.

Keywords:

Mathematics Subject Classification: Primary: 35Q40; Secondary: 35Q55, 35Q35, 82D37, 82C10.

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