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A Degenerate Fourth-Order Parabolic Equation Modeling Bose–Einstein Condensation. Part I: Local Existence of Solutions

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Abstract

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose–Einstein particles is analyzed. The model is a Fokker–Planck-type approximation of the Boltzmann–Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of a local-in-time nonnegative continuous weak solution is proven. If the solution is not global, it blows up with respect to the L ∞ norm in finite time. The proof is based on approximation arguments, interpolation inequalities in weighted Sobolev spaces, and suitable a priori estimates for a weighted gradient L 2 norm.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040, Wien, Austria

    Ansgar Jüngel

  2. Institut für Mathematik, Universität Paderborn, 33098, Paderborn, Germany

    Michael Winkler

Authors
  1. Ansgar Jüngel
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  2. Michael Winkler
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Correspondence to Ansgar Jüngel.

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Communicated by F. Otto

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Jüngel, A., Winkler, M. A Degenerate Fourth-Order Parabolic Equation Modeling Bose–Einstein Condensation. Part I: Local Existence of Solutions. Arch Rational Mech Anal 217, 935–973 (2015). https://doi.org/10.1007/s00205-015-0847-0

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  • DOI: https://doi.org/10.1007/s00205-015-0847-0

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