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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff

Pages: 187 - 212, Volume 24, Issue 1, May 2009      doi:10.3934/dcds.2009.24.187

 
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Yoshinori Morimoto - Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan (email)
Seiji Ukai - 17-26 Iwasaki, Hodogaya, Yokohama 240-0015, Japan (email)
Chao-Jiang Xu - Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France (email)
Tong Yang - Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China (email)

Abstract: Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case. As a further study on the problem in the spatially homogeneous situation, in this paper, we will prove the Gevrey smoothing property of the solutions to the Cauchy problem for Maxwellian molecules without angular cutoff by using pseudo-differential calculus. Furthermore, we apply similar analytic techniques for the Sobolev space regularity to the nonlinear equation, and prove the smoothing property of solutions for the spatially homogeneous nonlinear Boltzmann equation with the Debye-Yukawa potential.

Keywords:  Boltzmann equation, Debye-Yukawa potential, Gevrey hypoellipticity, non-cutoff cross-sections.
Mathematics Subject Classification:  Primary: 35B65, 35D05, 35D10, 35F20; Secondary: 76P05, 84C40.

Received: February 2007;      Revised: December 2007;      Available Online: January 2009.