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September  2008, 1(3): 453-489. doi: 10.3934/krm.2008.1.453

Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff

Zhaohui Huo 1, , Yoshinori Morimoto 2, , Seiji Ukai 3, and  Tong Yang 4,

1. 

Department of Mathematics, City University of Hong Kong, Hong Kong, China
2. 

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan
3. 

Liu Bie Ju Centre for Mathematical Sciences, City university of Hong Kong, Hong Kong, China
4. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  June 2008 Revised  June 2008 Published  August 2008

The spatially homogeneous Boltzmann equation without angular cutoff is discussed on the regularity of solutions for the modified hard potential and Debye-Yukawa potential. When the angular singularity of the cross section is moderate, any weak solution having the finite mass, energy and entropy lies in the Sobolev space of infinite order for any positive time, while for the general potentials, it lies in the Schwartz space if it has moments of arbitrary order. The main ingredients of the proof are the suitable choice of the mollifiers composed of pseudo-differential operators and the sharp estimates of the commutators of the Boltzmann collision operator and pseudo-differential operators. The method developed here also provides some new estimates on the collision operator.
Keywords: angular non-cutoff, commutator., regularity, Boltzmann equation, pseudo-differential operators.
Mathematics Subject Classification: 35B65, 35D10, 76P05, 82C4.
Citation: Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai, Tong Yang. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinetic & Related Models, 2008, 1 (3) : 453-489. doi: 10.3934/krm.2008.1.453
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