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Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

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Abstract

The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C ∞ regularity for any positive time.

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Authors and Affiliations

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

    Radjesvarane Alexandre

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

    Yoshinori Morimoto

  3. 17-26 Iwasaki-cho, Hodogaya-ku, Yokohama, 240-0015, Japan

    Seiji Ukai

  4. Université de Rouen, UMR 6085-CNRS Mathématiques, 76801, Saint-Etienne du Rouvray, France

    Chao-Jiang Xu

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

    Chao-Jiang Xu

  6. Department of Mathematics, City University of Hong Kong, Hong Kong, People’s Republic of China

    Tong Yang

Authors
  1. Radjesvarane Alexandre
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  2. Yoshinori Morimoto
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  3. Seiji Ukai
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  4. Chao-Jiang Xu
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  5. Tong Yang
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Corresponding author

Correspondence to Seiji Ukai.

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Communicated by T.-P. Liu

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Alexandre, R., Morimoto, Y., Ukai, S. et al. Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation. Arch Rational Mech Anal 198, 39–123 (2010). https://doi.org/10.1007/s00205-010-0290-1

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