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Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

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We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.

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Authors and Affiliations

  1. IRENAV Research Institute, French Naval Academy, Brest-Lanvéoc, 29290, France

    R. Alexandre

  2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China

    R. Alexandre

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

    Y. Morimoto

  4. 17-26 Iwasaki-cho, Hodogaya-ku, Yokohama, 240-0015, Japan

    S. Ukai

  5. School of Mathematics, Wuhan University, Wuhan, 430072, P. R. China

    C. -J. Xu

  6. Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801, Saint Etienne du Rouvray, France

    C. -J. Xu

  7. Department of mathematics, City University of Hong Kong, Hong Kong, P. R. China

    T. Yang

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  1. R. Alexandre
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  2. Y. Morimoto
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  3. S. Ukai
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  4. C. -J. Xu
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Correspondence to S. Ukai.

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Communicated by P. Constantin

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Alexandre, R., Morimoto, Y., Ukai, S. et al. Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff. Commun. Math. Phys. 304, 513–581 (2011). https://doi.org/10.1007/s00220-011-1242-9

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