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Analysis of the scattering operator for nonlinear dispersive equations

Research Project

Project/Area Number 25800074
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Mathematical analysis
Research InstitutionChiba University

Principal Investigator

Hironobu Sasaki  千葉大学, 大学院理学研究科, 准教授 (00568496)

Project Period (FY) 2013-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords非線型分散型方程式 / 波動作用素 / 散乱作用素 / 解析的平滑化効果 / 解の漸近挙動 / ソボレフ空間 / ローレンツ空間 / ベゾフ空間 / 非線型シュレーディンガー方程式 / 初期値問題 / 散乱問題 / 解析的な関数 / 斉次ソボレフ空間 / 非線形ディラック方程式 / 逆波動作用素 / 重み付ソボレフ空間 / 調和解析 / ディラック方程式 / 重み付きソボレフ空間
Outline of Final Research Achievements

In this study, we consider some problems on scattering operators and analytic smoothing effects for nonlinear dispersive equations. Using the Method of functional analysis, we obtain the following results:
(1) We considered the one space dimensional Dirac equation with a p-th power nonlinearity, and we proved that if p is greater than 3 then the scattering operator can be defined on a neighborhood of 0 in a suitable Hilbert space. (2) We considered the three space dimensional Dirac equation with a cubic power nonlinearity, and we proved that a neighborhood of 0 in the weighted Sobolev space is included in the same weighted Sobolev space. (3) We considered the Hartee equation, and we proved some properties of the analytic smoothing effect.

Report

(5 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (10 results)

All 2017 2016 2015 2014 Other

All Journal Article (6 results) (of which Peer Reviewed: 6 results,  Acknowledgement Compliant: 6 results) Presentation (2 results) Remarks (2 results)

  • [Journal Article] Remark on the scattering operator for the quintic nonlinear Dirac equation in one space dimension2017

    • Author(s)
      H. Sasaki
    • Journal Title

      Advanced Study of Pure Mathematics

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Remark on the analytic smoothing effect for the Hartree equation2017

    • Author(s)
      H. Sasaki
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Scattering operator for the one dimensional Dirac equation with power nonlinearity2016

    • Author(s)
      N. Hayashi and H. Sasaki
    • Journal Title

      J. Hyper. Differential Equations

      Volume: 13 Issue: 04 Pages: 821-832

    • DOI

      10.1142/s0219891616500223

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Small analytic solutions to the Hartree equation2016

    • Author(s)
      H. Sasaki
    • Journal Title

      J. Funct. Anal.

      Volume: 270 Issue: 3 Pages: 1064-1090

    • DOI

      10.1016/j.jfa.2015.11.001

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Remark on the scattering operator for the cubic nonlinear Dirac equation in three space dimensions2015

    • Author(s)
      H. Sasaki
    • Journal Title

      J. Differential Equations

      Volume: 259 Issue: 7 Pages: 3274-3297

    • DOI

      10.1016/j.jde.2015.04.020

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Small data scattering for the one-dimensional nonlinear Dirac equation with power nonlinearity2015

    • Author(s)
      H. Sasaki
    • Journal Title

      Comm. Partial Differential Equations

      Volume: 40 Issue: 11 Pages: 1959-2004

    • DOI

      10.1080/03605302.2015.1081608

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Small analytic solutions to the Hartree equation2016

    • Author(s)
      佐々木 浩宣
    • Organizer
      日本数学会2016年度年会函数方程式論分科会
    • Place of Presentation
      筑波大学(茨城県・つくば市)
    • Year and Date
      2016-03-19
    • Related Report
      2015 Research-status Report
  • [Presentation] Remark on the scattering operator for the cubic nonlinear Dirac equation in three space dimensions2014

    • Author(s)
      佐々木浩宣
    • Organizer
      日本数学会2014年度秋季総合分科会函数方程式論分科会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-28
    • Related Report
      2014 Research-status Report
  • [Remarks] 研究活動

    • URL

      http://www.math.s.chiba-u.ac.jp/~sasaki/J-re.html

    • Related Report
      2016 Annual Research Report 2015 Research-status Report 2014 Research-status Report
  • [Remarks] 研究活動

    • URL

      http://www.math.s.chiba-u.ac.jp/~sasaki/J-re.html

    • Related Report
      2013 Research-status Report

URL: 

Published: 2014-07-25   Modified: 2019-07-29  

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