Abstract
In this article, a simplified, hyperbolic model of the nonlinear, degenerate parabolic Kompaneets equation for the number density of photons is considered. It is shown that for non-negative, compactly supported initial data, weak solutions obeying a Kružkov entropy condition are unique. Other consequences for entropy solutions resulting from a contraction estimate are explored. Certain properties of entropy solutions are investigated and convergence in time of entropy solutions with compactly supported initial data to stationary solutions is shown. The development of a Bose–Einstein condensate for initial data under certain conditions is proven. It is also shown that the total number of photons not in a Bose–Einstein condensate is non-increasing in time, and that any such loss of photons is only to the condensate.
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Acknowledgements
J. B. acknowledges support from the National Science Foundation under the grant DMS-1401732 and from the Center of Nonlinear Analysis at Carnegie Mellon University.
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Ballew, J. (2018). Bose–Einstein Condensation and Global Dynamics of Solutions to a Hyperbolic Kompaneets Equation. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 236. Springer, Cham. https://doi.org/10.1007/978-3-319-91545-6_9
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