Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 63–89 | Cite as

Uniform in Time Lower Bound for Solutions to a Quantum Boltzmann Equation of Bosons

  • Toan T. NguyenEmail author
  • Minh-Binh Tran


In this paper, we consider a quantum Boltzmann equation, which describes the interaction between excited atoms and a condensate. The collision integrals are taken–over energy manifolds, having the full form of the Bogoliubov dispersion law for particle energy. We prove that nonnegative radially symmetric solutions of the quantum Boltzmann equation are bounded from below by a Gaussian distribution, uniformly in time.


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The authors thank the referees for their constructive comments. M.-B Tran has been supported by NSF Grant DMS-1814149, NSF Grant RNMS (Ki-Net) 1107291. M.-B Tran would like to thank Professor Daniel Heinzen, Professor Linda Reichl, Professor Mark Raizen and Professor Robert Dorfman for fruitful discussions on the topic. The research was carried on while M.-B. Tran was visiting University of Texas at Austin and Penn State University. He would like to thank these institutions for their hospitality.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityState CollegeUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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