Kinetic and Related Models (KRM)

Bounded solutions of the Boltzmann equation in the whole space

Pages: 17 - 40, Volume 4, Issue 1, March 2011      doi:10.3934/krm.2011.4.17

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Radjesvarane Alexandre - Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China (email)
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 - School of Mathematics, Wuhan University, 430072, Wuhan, China (email)
Tong Yang - Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China (email)

Abstract: We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.

Keywords:  Boltzmann equation, local existence, locally uniform Sobolev space, spatial behavior at infinity, pseudo-differential calculus.
Mathematics Subject Classification:  Primary: 35A01, 35A02, 35A09, 35S05; Secondary: 76P05, 82C40.

Received: October 2010;      Revised: October 2010;      Available Online: January 2011.