Eternal solutions of the Boltzmann equation near travelling Maxwellians

It is shown in this paper that the Cauchy problem of the Boltzmann equation, with a cut-off soft potential and an initial datum close to a travelling Maxwellian, has a unique positive eternal solution. This eternal solution is exponentially decreasing at infinity for all t ∈ ( − ∞ , ∞ ) , consequent...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2006-02, Vol.314 (1), p.219-232
Main Authors: Wei, Jinbo, Zhang, Xianwen
Format: Article
Language:English
Subjects:
Boltzmann equation
Eternal positive solution
Exact sciences and technology
Kaniel–Shinbrot iteration
Mathematical analysis
Mathematics
Partial differential equations
Sciences and techniques of general use
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Summary:It is shown in this paper that the Cauchy problem of the Boltzmann equation, with a cut-off soft potential and an initial datum close to a travelling Maxwellian, has a unique positive eternal solution. This eternal solution is exponentially decreasing at infinity for all t ∈ ( − ∞ , ∞ ) , consequently the moments of any order are finite. This result gives a negative answer to the conjecture of Villani in the spatially inhomogeneous case.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2005.03.080