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半相対論的非線形場の数学的基礎

Research Project

Project/Area Number 16J30008
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Mathematical analysis
Research InstitutionWaseda University

Principal Investigator

藤原 和将  早稲田大学, 理工学術院, 特別研究員(PD)

Project Period (FY) 2016-04-22 – 2018-03-31
Project Status Discontinued (Fiscal Year 2017)
Budget Amount *help
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥900,000 (Direct Cost: ¥900,000)
Keywords半相対論的方程式 / 時間大域的可解性 / 冪乗型非線型項 / 臨界尺度 / 初期値問題の適切性 / 初期値問題の解の有限時刻爆発
Outline of Annual Research Achievements

29年度の研究では、ゲージ不変性を伴わない冪乗型の非線型項を有する半相対論的方程式の初期値問題に対して、時間大域的な解が存在しない為の条件を研究した。

ゲージ不変性を伴わない冪乗型の非線型項を有する方程式の初期値問題を考察した場合、適当な条件の下で時間大域的に解が存在しない事が期待される。一方で、半相対論的方程式は分数階の微分作用素を主要部に有する為、従来の手法で解の延長不能性に就いて議論する事が出来ない。報告者と受け入れ研究員は、分数階微分を主要部から取り除く為の半相対論的方程式の変形を考案する事で、解が適当な条件の下、時間大域的に存在しない事を示した。その後この変形による半相対論的方程式の考察は、戍亥隆恭氏によって尺度不変性の観点から自然なものに拡張されている。

29年度の研究では、適当な試験関数に対する分数階微分を具体的に計算する事で、方程式を変形することなく、より直接的な手法で時間大域的な解の非存在を示した。分数階微分は非局所的な作用素であり、分数階の導関数を、元の関数を用いて各点で制御する事は一般に困難である。特に、試験関数としてよく用いられる台がコンパクトな関数に対しては、分数階の導関数の各点に於ける評価は不可能であるが、本研究では減衰する多項式に対する分数階の導関数の各点に対する評価を具体的に計算し、尺度劣臨界に対する解の時間大域可解性をより直接的な方法によって否定した。特に、分数階の導関数の空間遠方に対する減衰の速度が、多項式の冪によって、対応する古典的な導関数の速度と同様になる場合と、遅くなる場合がある事を示した。

Research Progress Status

29年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

29年度が最終年度であるため、記入しない。

Report

(2 results)
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • Research Products

    (18 results)

All 2017 2016 Other

All Journal Article (7 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 7 results,  Acknowledgement Compliant: 4 results,  Open Access: 1 results) Presentation (10 results) (of which Int'l Joint Research: 5 results,  Invited: 6 results) Remarks (1 results)

  • [Journal Article] Lifespan of strong solutions to the periodic nonlinear Schroedinger equation without gauge invariance2017

    • Author(s)
      K. Fujiwara, T. Ozawa
    • Journal Title

      Journal of Evolution Equations

      Volume: 印刷中 Issue: 3 Pages: 1023-1030

    • DOI

      10.1007/s00028-016-0364-0

    • Related Report
      2017 Annual Research Report 2016 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Higher order fractional Leibniz rule2017

    • Author(s)
      Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa
    • Journal Title

      Journal of Fourier Analysis and Applications

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Blow-up of solutions for weakly coupled systems of complex Ginzburg-Landau equations2017

    • Author(s)
      Kazumasa Fujiwara, Masahiro Ikeda, Yuta Wakasugi
    • Journal Title

      Electronic Journal of Differential Equations

      Volume: 17

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity2017

    • Author(s)
      Kazumasa Fujiwara, Masahiro Ikeda, Yuta Wakasugi
    • Journal Title

      Funkcialaj Ekvacioj

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Blow-up for self-interacting fractional Ginzburg-Landau equation2017

    • Author(s)
      Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa
    • Journal Title

      Dynamics of PDE

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Finite time blowup of solutions to the nonlinear Schroedinger equation without gauge invariance2016

    • Author(s)
      K. Fujiwara, T. Ozawa
    • Journal Title

      Journal of Mathematical Physics

      Volume: 57 Issue: 8 Pages: 082130-082130

    • DOI

      10.1063/1.4960725

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Weighted <i>L</i><sup><i>p</i></sup>-boundedness of convolution type integral operators associated with bilinear estimates in the Sobolev spaces2016

    • Author(s)
      K. Fujiwara, T. Ozawa
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 68 Issue: 1 Pages: 169-191

    • DOI

      10.2969/jmsj/06810169

    • NAID

      130005122687

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Higher order fractional Leibniz estimates2017

    • Author(s)
      Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa
    • Organizer
      ISAAC Congress 2017
    • Place of Presentation
      Linnaeus University
    • Year and Date
      2017-08-14
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Lifespan of strong solutions to the periodic derivative nonlinear Schro:dinger equation2017

    • Author(s)
      Kazumasa Fujiwara, Tohru Ozawa
    • Organizer
      International Workshop on “Fundamental Problems in Mathematical and Theoretical Physics”
    • Place of Presentation
      早稲田大学
    • Year and Date
      2017-07-24
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Blow-up for self-interacting fractional Ginzburg-Landau equation2017

    • Author(s)
      K. Fujiwara, V. Georgiev, and T. Ozawa
    • Organizer
      第 24 回 応用解析研究会 シンポジウム
    • Place of Presentation
      箱根湯本温泉 ホテル「明日香」
    • Year and Date
      2017-03-13
    • Related Report
      2016 Annual Research Report
  • [Presentation] Blow-up for self-interacting fractional Ginzburg-Landau equation and related commutator estimates2017

    • Author(s)
      Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa
    • Organizer
      熊本大学応用解析セミナー
    • Place of Presentation
      熊本大学
    • Related Report
      2017 Annual Research Report 2016 Annual Research Report
    • Invited
  • [Presentation] An ODE approach for blow-up of some evolution equations in Fujita subcritical case2017

    • Author(s)
      Kazumasa Fujiwara, Masahiro Ikeda, Tohru Ozawa, Yuta Wakasugi
    • Organizer
      大阪駅前セミナー
    • Place of Presentation
      龍谷大学
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] An ODE approach for the blow-up for nonlinear damped wave equations with time dependent damping2017

    • Author(s)
      Kazumasa Fujiwara, Masahiro Ikeda, Yuta Wakasugi
    • Organizer
      なかもず解析セミナー
    • Place of Presentation
      大阪府立大学
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] A remark on blowup for weakly coupled heat systems by an ODE approach2017

    • Author(s)
      Kazumasa Fujiwara, Masahiro Ikeda, Yuta Wakasugi
    • Organizer
      EDP e DINTORNI
    • Place of Presentation
      Universita` di Bari
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Blow-up for self-interacting fractional Ginzburg-Landau equation2017

    • Author(s)
      K. Fujiwara, V. Georgiev, and T. Ozawa
    • Organizer
      中央大学偏微分方程式セミナー
    • Place of Presentation
      中央大学
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Higher order fractional Leibniz rule2016

    • Author(s)
      K. Fujiwara, V. Georgiev, and T. Ozawa
    • Organizer
      Mathematical Analysis for Stability in Nonlinear Dynamics -in honor of Professor Vladimir Georgiev on his 60th birthday
    • Place of Presentation
      北海道大学
    • Year and Date
      2016-08-24
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Lifespan of strong solutions to the periodic nonlinear Schrodinger equation without gauge invariance2016

    • Author(s)
      K. Fujiwara and T. Ozawa
    • Organizer
      11th AIMS Conference Series on Dynamical Systems and Differential Equations
    • Place of Presentation
      Hyatt Regency Orlando
    • Year and Date
      2016-07-01
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research
  • [Remarks] 藤原和将のホームページ

    • URL

      http://kfujiwara.webcrow.jp/home_j.html

    • Related Report
      2017 Annual Research Report

URL: 

Published: 2016-05-17   Modified: 2024-03-26  

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