Bounded  solutions of the Boltzmann equationin the whole space

Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang

doi:10.3934/krm.2011.4.17

Article Contents

2011, Volume 4, Issue 1: 17-40. Doi: 10.3934/krm.2011.4.17

This issue Previous Article Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type Next Article Gain of integrability for the Boltzmann collisional operator

Bounded solutions of the Boltzmann equation in the whole space

1.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240
2.
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501
3.
17-26 Iwasaki, Hodogaya, Yokohama 240-0015
4.
School of Mathematics, Wuhan University, 430072, Wuhan
5.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received: September 30, 2010

Revised: September 30, 2010

Published: February 2011

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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.

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Mathematics Subject Classification: Primary: 35A01, 35A02, 35A09, 35S05; Secondary: 76P05, 82C40.

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