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A study on an estimate of solutions of Helmholtz equation and the smoothing effect of solutions of corresponding time-dependent problems

Research Project

Project/Area Number 16K05243
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionNippon Medical School

Principal Investigator

Nakazawa Hideo  日本医科大学, 医学部, 教授 (80383371)

Co-Investigator(Kenkyū-buntansha) 門脇 光輝  滋賀県立大学, 工学部, 教授 (70300548)
望月 清  東京都立大学, 理学研究科, 客員教授 (80026773)
渡辺 一雄 (渡邊 一雄)  学習院大学, 理学部, 講師 (90260851)
Project Period (FY) 2016-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords波動方程式 / シュレディンガー作用素 / ヘルムホルツ方程式 / 非自己共役 / スペクトル解析 / 数学的散乱理論 / 一様リゾルベント評価 / 極限振幅の原理 / 摩擦項 / リゾルベント評価 / 非自己共役作用素 / 消散作用素 / スペクトル理論 / 極限吸収の原理 / シュレーディンガー方程式 / 偏微分方程式 / 解の挙動 / 定常問題 / 散乱問題 / 極限吸収原理 / 外部問題 / Hardyの不等式 / 散乱の定常問題 / 平滑化評価 / Strichartz評価 / 偏微分方程式論
Outline of Final Research Achievements

In this study, we have derived a uniform estimate with respect to spectral parameters and related estimates for the solution of the Helmholtz equation with energy-dependent potential, which is a stationary problem for the wave equation with dissipative term, and studied the behavior of the solution of the original wave equation as its application. As a result of this work, a new estimate of the solution to the stationary problem was derived, and the principle of the limiting amplitude for the wave equation with dissipative term, which also takes into account the effect of magnetic fields, was successfully proved. This is an improvement of the result obtained by Mizohata-Mochizuki in 1966 (J. Math. Kyoto Univ.,). This result has already been submitted to a peer-reviewed journal and has been accepted for publication.

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Academic Significance and Societal Importance of the Research Achievements

本研究では,1966年以来全く進展のなかった,摩擦項を伴う波動方程式に対する極限振幅の原理を,3次元全空間のみならず3以上の全ての次元に対する全空間或いは2以上の全ての次元に対する星状な障害物の外部領域として,更に磁場の効果をも取り入れた形で証明することに成功した。その証明で重要な役割を果たす評価は,Mochizuki(2010, RIMS)及びMochizuki-Nakazawa(2015,RIMS)によって得られた磁場中のシュレディンガー作用素に対する一様リゾルベント評価である。これらの評価によって外部領域における定常問題の解の評価が可能となり,今回の結果が得られた。

Report

(8 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
  • 2016 Research-status Report
  • Research Products

    (13 results)

All 2023 2022 2019 2018 2017 2016 Other

All Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Acknowledgement Compliant: 1 results) Presentation (6 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results) Remarks (2 results) Funded Workshop (2 results)

  • [Journal Article] A Uniform Resolvent Estimate for a Helmholtz Equation with Some Large Perturbations in An Exterior Domain2022

    • Author(s)
      Hideo Nakazawa
    • Journal Title

      Current Trends in Analysis, its Applications and Computation. Trends in Mathematics

      Volume: - Pages: 633-641

    • DOI

      10.1007/978-3-030-87502-2_63

    • ISBN
      9783030875015, 9783030875022
    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Uniform resolvent estimates for stationary dissipative wave equations in an exterior domain and their application to the principle of limiting amplitude.2017

    • Author(s)
      Kiyoshi Mochizuki, Hideo Nakazawa
    • Journal Title

      New Trends in Analysis and Interdisciplinary Applications: Selected Contributions of the 10th ISAAC Congress, Macau 2015 (Trends in Mathematics), Birkhauser

      Volume: 1 Pages: 521-527

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Uniform Resolvent Estimates for Stationary DissipativeWave Equations in an Exterior Domain and Their Application to the Principle of Limiting Amplitude2017

    • Author(s)
      Kiyoshi Mochizuki and Hideo Nakazawa
    • Journal Title

      New Trends in Analysis and Interdisciplinary Applications: Selected Contributions of the 10th ISAAC Congress, Macau 2015 (Trends in Mathematics), Birkhaeuser

      Volume: 1 Pages: 521-527

    • DOI

      10.1007/978-3-319-48812-7_66

    • ISBN
      9783319488103, 9783319488127
    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] 摩擦項をもつ磁場中の波動方程式に対する極限振幅の原理2023

    • Author(s)
      中澤秀夫
    • Organizer
      日本数学会 (函数解析学分科会)
    • Related Report
      2022 Annual Research Report
  • [Presentation] 摩擦項をもつ磁場中の波動方程式に対する極限振幅の原理2022

    • Author(s)
      中澤秀夫
    • Organizer
      駿河台偏微分方程式研究集会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Some estimates of solutions of perturbed Helmholtz equations2019

    • Author(s)
      Hideo Nakazawa
    • Organizer
      ISAAC 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] 大きな摂動項を伴うヘルムホルツ方程式の一様リゾルベント評価とその応用2019

    • Author(s)
      中澤秀夫
    • Organizer
      つくば偏微分方程式研究集会
    • Related Report
      2019 Research-status Report
  • [Presentation] ヘルムホルツ方程式の解の評価2019

    • Author(s)
      中澤秀夫
    • Organizer
      名古屋偏微分方程式研究集会
    • Related Report
      2018 Research-status Report
  • [Presentation] 磁場中のシュレディンガー方程式に対する 一様リゾルベント評価とその応用I&II2016

    • Author(s)
      中澤秀夫
    • Organizer
      平成28 年度RIMS共同研究 「微分方程式に対する散乱理論の展開」
    • Place of Presentation
      京都大学数理解析研究所111号室
    • Year and Date
      2016-09-03
    • Related Report
      2016 Research-status Report
    • Invited
  • [Remarks] researchmap 中澤秀夫(Hideo Nakazawa)

    • URL

      https://researchmap.jp/hideo_nakazawa_nms

    • Related Report
      2022 Annual Research Report
  • [Remarks] 平成 28 年度 RIMS 共同研究 「微分方程式に対する散乱理論の展開」

    • URL

      http://www.kurims.kyoto-u.ac.jp/~kyodo/kaizuka20160729j.pdf

    • Related Report
      2016 Research-status Report
  • [Funded Workshop] Himeji Conference on Partial Differential Equations2018

    • Related Report
      2017 Research-status Report
  • [Funded Workshop] Tosio Kato Centennial Conference2017

    • Related Report
      2017 Research-status Report

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Published: 2016-04-21   Modified: 2024-01-30  

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