On a Boltzmann Equation for Compton Scattering from Non relativistic Electrons at Low Density
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A Boltzmann equation, used to describe the evolution of the density function of a gas of photons interacting by Compton scattering with electrons at low density and non relativistic equilibrium, is considered. A truncation of the very singular redistribution function is introduced and justified. The existence of weak solutions is proved for a large set of initial data. A simplified equation, where only the quadratic terms are kept and that appears at very low temperature of the electron gas, for small values of the photon’s energies, is also studied. The existence of weak solutions, and also of more regular solutions that are very flat near the origin, is proved. The long time asymptotic behavior of weak solutions of the simplified equation is described.
KeywordsKinetic equation Compton scattering Low density non relativistic electrons Kompaneets equation Weak solutions Long time behavior Dirac measures
The research of the first author is supported by the Basque Government through the BERC 2014-2017 program, by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323, and by MTM2014-52347-C2-1-R of DGES. The research of the second author is supported by Grants MTM2014-52347-C2-1-R of DGES and IT641-13 of the Basque Government. The authors gratefully thank the referees for their careful reading of the manuscript and their valuable comments and recommendations.
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