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Letters in Mathematical Physics

, Volume 108, Issue 10, pp 2255–2283 | Cite as

Uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov equations in \(\mathbb {R}^{1+1}\)

  • Jacky Jia Wei Chong
Article
  • 31 Downloads

Abstract

We prove the global well-posedness of the time-dependent Hartree–Fock–Bogoliubov (TDHFB) equations in \(\mathbb {R}^{1+1}\) with two-body interaction potential of the form \(N^{-1}v_N(x) = N^{\beta -1} v(N^\beta x)\) where \(v\ge 0\) is a sufficiently regular radial function, i.e., \(v \in L^1(\mathbb {R})\cap C^\infty (\mathbb {R})\). In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon (Commun Partial Differ Equ 42:24–67, 2017), we are able to show for any scaling parameter \(\beta >0\) the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of N, cf. (Bach et al. in The time-dependent Hartree–Fock–Bogoliubov equations for Bosons, 2016. arXiv:1602.05171).

Keywords

Mean-field Hartree-Fock-Bogoliubov equations Global well-posedness 

Mathematics Subject Classification

35Q40 35Q55 81V70 82C10 

Notes

Acknowledgements

The author would like to take this opportunity to express his deepest gratitude toward his two advisors M. Grillakis and M. Machedon for all the time and energy they have spent on the author. Moreover, the author would also like to thank the editors and referees for providing valuable feedbacks. In particular, one of the referees has the author’s sincere appreciation for submitting a very detailed and thoughtful review of the article, which significantly contributed to improving the presentation quality of the paper.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

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