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Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

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Abstract

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad’s angular cutoff assumption. More precisely, for solutions inside the finite Mach number region (time like region), we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region (space like region), we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results extend the classical results of Liu and Yu (Commun Pure Appl Math 57:1543–1608, 2004), (Bull Inst Math Acad Sin 1:1–78, 2006), (Bull Inst Math Acad Sin 6:151–243, 2011) and Lee et al. (Commun Math Phys 269:17–37, 2007) to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.

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Authors and Affiliations

  1. Department of Mathematics, National Cheng Kung University, Tainan, Taiwan

    Yu-Chu Lin & Kung-Chien Wu

  2. Institute of Natural Sciences and School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China

    Haitao Wang

  3. National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan

    Kung-Chien Wu

Authors
  1. Yu-Chu Lin
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  2. Haitao Wang
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  3. Kung-Chien Wu
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Correspondence to Kung-Chien Wu.

Additional information

Yu-Chu Lin is supported by the Ministry of Science and Technology under the Grant MOST 105-2115-M-006-002-. Haitao Wang is sponsored by Shanghai Sailing Program (18YF1411800). Kung-Chien Wu is supported by the Ministry of Science and Technology under the Grant 104-2628-M-006-003-MY4 and National Center for Theoretical Sciences.

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Lin, YC., Wang, H. & Wu, KC. Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation. J Stat Phys 171, 927–964 (2018). https://doi.org/10.1007/s10955-018-2047-4

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  • DOI: https://doi.org/10.1007/s10955-018-2047-4

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