Abstract
This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming S n−1 integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel–Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes the large data near local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in (Alonso et al. in Convolution inequalities for the Boltzmann collision operator. arXiv:0902.0507 [math.AP]) , by E. Carneiro and the authors, that allows us to show such propagation without additional conditions on the collision kernel. Finally, an L p-stability result (with 1≤p≤∞) is presented assuming the aforementioned condition.
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The authors acknowledge partial support from NSF grants DMS-0636586 and DMS 0807712. Support from the Institute from Computational Engineering and Sciences at the University of Texas at Austin is also gratefully acknowledged.
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Alonso, R.J., Gamba, I.M. Distributional and Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials with Integrable Angular Cross Section. J Stat Phys 137, 1147–1165 (2009). https://doi.org/10.1007/s10955-009-9873-3
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DOI: https://doi.org/10.1007/s10955-009-9873-3