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Nonlinear microlocal analysis for the Boltzmann equation

Research Project

Project/Area Number 17K05318
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionKyoto University

Principal Investigator

Morimoto Yoshinori  京都大学, 人間・環境学研究科, 名誉教授 (30115646)

Co-Investigator(Kenkyū-buntansha) 清水 扇丈  京都大学, 人間・環境学研究科, 教授 (50273165)
Project Period (FY) 2017-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywordsボルツマン方程式 / 非切断型 / 多項式減衰の摂動解 / ランダウ方程式 / 解析的平滑化効果 / 指数型積分演算子 / 対数オーダー演算子 / 時間大域解 / 指数型フーリエ演算子 / 対数オーダー積分型演算子 / 平滑効果 / 準楕円性 / 無限次退化 / Debyeー湯川型ポテンシャル / Debye-湯川型ポテンシャル / 初期値問題 / 空間非一様 / 解析的平滑効果 / 多項式オーダー摂動解 / 解の平滑効果 / 線形化作用素 / 非線形超局所解析 / 解の一意性 / 切断近似
Outline of Final Research Achievements

The non-cutoff Boltzmann equation and the Landau equation related as its singular limit are kinetic equations in consideration of a long-range interaction of particles. The time global existence and the smoothing effect of solutions to the Cauchy problem for those equations are studied by using the Fourier multiplier of the integral form.
For the non-cutoff Boltzmann equation,we obtained the time global solution perturbed around the global equilibrium which decays polynomial order with respect to the velocity variable. As to the Landau equation, it was shown for the simple Maxwellian case that the analytic smoothing effect of the perturbation around the global equilibrium occurs if the perturbation decays with the square root of the equilibrium.

Academic Significance and Societal Importance of the Research Achievements

積分型フーリエ乗法演算子はボルツマン方程式の輸送項と衝突項から、空間変数の楕円性を自然に導く性質を持っている。楕円性の指数は衝突項の速度変数に関する楕円性の指数に等しく、この指数はボルツマン衝突積分項の角度変数の特異性の指数 2s (s<1) に一致する。それゆえ非切断型ボルツマン方程式の初期値問題の解については 指数 min{2s,1}の Gevrey 級での平滑効果が期待されていた。この予想の解決の第一歩として、s=1 とした極限の場合として考えられるランダウ方程式(衝突項が速度変数に関する非線形2階偏微分作用素)の簡単なモデルについて、初めて解析的平滑効果の証明に成功した。

Report

(7 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (32 results)

All 2022 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (16 results) Journal Article (4 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 4 results) Presentation (12 results) (of which Int'l Joint Research: 6 results,  Invited: 10 results)

  • [Int'l Joint Research] 中国/南京航空航天大学/City University of Hong Kong(中国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Simon Fraser University(カナダ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] 南京航空航天大学/City University of Hong Kong(中国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] PUC-Rio(ブラジル)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] University of Rouen-Normandie(フランス)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] 南京航空航天大学/City University of Hong Kong(中国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Simon Fraser University(カナダ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] PUC-Rio(ブラジル)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Rouen University(フランス)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] PUC-Rio(ブラジル)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] City University of Hong Kong(中国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Simon Frase University(カナダ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Rouen University(フランス)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] City University of Hong Kong(中国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Simon Fraser University(カナダ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] PUC Rio(ブラジル)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Non-cutoff Boltzmann equation with polynomial decay perturbations.2021

    • Author(s)
      R. Alonso, Y. Morimoto, W. Sun, T. Yang
    • Journal Title

      Rev. Mat. Iberoam.

      Volume: 37 Issue: 1 Pages: 189-292

    • DOI

      10.4171/rmi/1206

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Non-cutoff Boltzmann Equation with Polynomial Decay Perturbations2021

    • Author(s)
      Ricardo Alonso, Yoshinori Morimoto, Weiran Sun, Tong Yang
    • Journal Title

      Revista Matematica Iberoamericana

      Volume: 37.1

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules.2020

    • Author(s)
      Y. Morimoto, C.-J. Xu
    • Journal Title

      Kinet. Relat. Models

      Volume: 13 Issue: 5 Pages: 951-978

    • DOI

      10.3934/krm.2020033

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules2020

    • Author(s)
      Yoshinori Morimoto, Chao-Jiang Xu
    • Journal Title

      Kinetic and Related Models

      Volume: 未定

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 対数オーダー積分型演算子とその応用2022

    • Author(s)
      森本 芳則
    • Organizer
      第38回松山キャンプ
    • Related Report
      2022 Annual Research Report
  • [Presentation] The Boltzmann equation for Debye-Yukawa potential2020

    • Author(s)
      森本 芳則
    • Organizer
      第36回 松山キャンプ
    • Related Report
      2020 Research-status Report
  • [Presentation] Analytic smoothing effect for the full Landau equation of Maxwellian molecules2019

    • Author(s)
      Yoshinori Morimoto
    • Organizer
      2019 CAU-RIMS Joint Workshop on Nonlinear PDEs, Korea
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Ukai inequality and its application to kinetic equations2019

    • Author(s)
      Yoshinori Morimoto
    • Organizer
      2019 南航偏微分方程青年学者会議, 南京航空航天大学
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Kolmogorov型方程式の基本解とその運動論方程式への応用2019

    • Author(s)
      森本芳則
    • Organizer
      茨城大学理学部数学教室 第41 回金曜解析セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Fundamental solution for Kolmogorov type equation and its application to the kinetic equation2019

    • Author(s)
      Yoshinori Morimoto
    • Organizer
      Chongqing University, China
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 非切断型ボルツマン方程式の多項式減衰な摂動大域解2018

    • Author(s)
      Yoshinori Morimoto
    • Organizer
      東北大学理学研究科数学専攻、談話会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Analytic smoothing effect for the full Landau equation of Maxwellian molecules2018

    • Author(s)
      Yoshinori Morimoto
    • Organizer
      2019 CAU-RIMS Joint Workshop on Nonlinear PDEs
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The spatially homogeneous Boltzmann Equation for Debye-Yukawa potential2017

    • Author(s)
      Yoshinori Morimoto
    • Organizer
      International Conference on Nonlinear Analysis: Kinetic Theory, Gas Dynamics, and Related Fields. Institute of Mathematics, Academia Sinica, Taipei, Taiwan
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Revisit on the spatially homogeneous Boltzmann equation for Debye-Yukawa potential2017

    • Author(s)
      森本芳則
    • Organizer
      京都大学 NLPDE セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Revisit on the Boltzmann equation for Debye-Yukawa potential2017

    • Author(s)
      Yoshinori Morimoto
    • Organizer
      Workshop on Hyperbolic and Parabolic Systems, Mathematics and Physics Unit“ Multiscale Analysis, Modelling and Simulation ” Top Global University Project, Waseda University
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 切断近似をしないボルツマン方程式の時間大域解2017

    • Author(s)
      森本芳則
    • Organizer
      Workshop on Transport Theory and Biomedical Optics: 輸送理論と生体医用光学 産総研:臨海副都心センター
    • Related Report
      2017 Research-status Report
    • Invited

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Published: 2017-04-28   Modified: 2024-01-30  

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