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非局所相互作用系と完全可積分構造の関連の探索

Research Project

Project/Area Number 19654028
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeSingle-year Grants
Research Field Global analysis
Research InstitutionTohoku University

Principal Investigator

小川 卓克  Tohoku University, 大学院・理学研究科, 教授 (20224107)

Co-Investigator(Kenkyū-buntansha) 永井 敏隆  広島大学, 大学院・理学研究科, 教授 (40112172)
加藤 圭一  東京理科大学, 理学部一部, 准教授 (50224499)
石渡 通徳  室蘭工業大学, 工学部, 准教授 (30350458)
Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2009: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2008: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2007: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywords非局所放物型方程式 / 臨界密度 / 半導体シュミレーション / 走化性粘菌モデル / 移流拡散方程式 / 臨界空間 / 自己相似解 / 非線形熱方程式 / 非局所方程式 / 半導体素子設計 / 非線形偏微分方程式 / 完全可積分系 / 関数方程式論 / 非線形波動方程式
Research Abstract

研究実績は以下のとおり。
1.研究代表者の小川は、連携研究者の石渡通徳(室蘭工大・工)、研究協力者の高橋太(阪市立大・理)、黒木場正城(福岡大・理)らとともに2次元上で楕円型部分が非線形に摂動されたKeller-Segel系の時間適切性について研究し、初期値が十分小さく解に一定の変分法的特性を満たす場合には解が一意的に存在すること、またそれ以外の場合には一般に解の一意性が崩れ、適切性が成立しないことを変分的手法により示した。特に初期値問題が非適切であって少なくとも球対称の初期値からは少なくとも解が2つ存在することを示した。方法はヒルベルト-シュミット法と楕円型方程式の臨界点解からの分岐解をとらえることにより示される。
2.研究代表者の小川は研究協力者の清水扇丈(静岡大・理)と共同で2次元Drift-diffusion方程式を臨界Besov空間で考え、局所解の存在定理と時間大域的可解性を示した。その際に非回帰的Banach空間における最大正則性定理を証明し、L^1に近い空間における擬似的なエネルギー不等式が成り立つこと、またL^1空間では同様の不等式が一般には成立しないことを示した。
3.研究代表者の小川は研究協力者の山本征法(東北大大学院博士3年)と共同で、高次元drift-diffusion方程式の解の減衰について研究し、時間大域解の解の漸近挙動を高次の項まで展開した。特にこの問題に固有のキャンセル効果により高次漸近展開項がより簡潔に表せることと、高次の展開項が一般には消えないことを示し、高次項の誤差項に対する下からの減衰評価を与えた。

Report

(3 results)
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (25 results)

All 2009 2008 2007

All Journal Article (14 results) (of which Peer Reviewed: 14 results) Presentation (8 results) Book (3 results)

  • [Journal Article] Multiple global existence of solutions for nonlinearly perturbed elliptic parabolic system in R^22009

    • Author(s)
      T.Ogawa
    • Journal Title

      Electric J.Differential Equations 2009

      Pages: 1-10

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Non-existence of weak solutions to nonlinear damped wave equations in exterior domains2009

    • Author(s)
      T.Ogawa
    • Journal Title

      Nonlinear Anal 70

      Pages: 3696-3701

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic bahavior of solutions to drift-diffution system with generalized dissipation2009

    • Author(s)
      T.Ogawa
    • Journal Title

      Math.Models Methods Appl.Sci. 19

      Pages: 939-967

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global existence of solutions for a nonlinearly perturbed elliptic parabolic System in R^22009

    • Author(s)
      T.Ogawa
    • Journal Title

      Z.Angew.Math.Phys. 60

      Pages: 840-867

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] End-point maximal regularity and wellposedness of the two dimensional Keller-Segel system in a critical Besov space2009

    • Author(s)
      T.Ogawa
    • Journal Title

      Math.Z. 264

      Pages: 601-628

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Well-posedness for the drift-diffusion system in L^p arisinq from the semiconductor device simulation2008

    • Author(s)
      T. Ogawa
    • Journal Title

      J. Math. Anal. Appl 342

      Pages: 1052-1067

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The drift-diffusion system in two dimensional critical Hardy space2008

    • Author(s)
      T. Ogawa
    • Journal Title

      J. Funct Anal 255

      Pages: 1107-1138

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Regularity condition by mean oscillation to a weak solution of the harmonic heat flow into sphere2008

    • Author(s)
      T. Ogawa
    • Journal Title

      Cal. Val. in P.D.E 33

      Pages: 391-415

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type2008

    • Author(s)
      T. Ogawa
    • Journal Title

      Integral Diff. Equations 21

      Pages: 1113-1154

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Well-posedness for the drift-diffusion system in Lp arising from the semi-conductor device simulation,2008

    • Author(s)
      M. Kurokiba and T. Ogawa,
    • Journal Title

      Comm. Pure Appl. Anal. 342

      Pages: 1052-1067

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Regularity condition by mean oscillation to a weak solution of the har-monic heat flow into sphere,2008

    • Author(s)
      M. Misawa and T. Ogawa
    • Journal Title

      Cal. Val. Partial Diff. Equations on line

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space2007

    • Author(s)
      T. Nagai and T. Yamada,
    • Journal Title

      J. Math Anal. Appl, 336

      Pages: 704-726

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On bounds for global solutions of semilinear parabolic equations with critical and subcritical Sobolev exponent,2007

    • Author(s)
      M. Ishiwata,
    • Journal Title

      Differetinal Integral Equations, 20

      Pages: 1021-1034

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Multiple solutions for semilinear elliptic systems involving critical Sobolev exponent,2007

    • Author(s)
      M. Ishiwata,
    • Journal Title

      Differetinal Integral Equations, 20

      Pages: 1237-1252

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Drift-diffusion system in critical case2009

    • Author(s)
      T.Ogawa
    • Organizer
      Harmonic Analysis and Partial Differential Equations
    • Place of Presentation
      名古屋
    • Year and Date
      2009-10-02
    • Related Report
      2009 Annual Research Report
  • [Presentation] Drift-diffusion system in critical case2009

    • Author(s)
      T.Ogawa
    • Organizer
      Japan-German PDE seminar
    • Place of Presentation
      Helmholz center(ドイツ)
    • Year and Date
      2009-09-10
    • Related Report
      2009 Annual Research Report
  • [Presentation] Drift-diffusion system in the critical space2008

    • Author(s)
      小川卓克
    • Organizer
      北九州における偏微分方程式work shop
    • Place of Presentation
      九州工業大学
    • Year and Date
      2008-11-29
    • Related Report
      2008 Annual Research Report
  • [Presentation] Drift-diffusion system in the critical space2008

    • Author(s)
      小川卓克
    • Organizer
      Nonlinear PDE Workshop at Sendai
    • Place of Presentation
      東北大学
    • Year and Date
      2008-11-28
    • Related Report
      2008 Annual Research Report
  • [Presentation] Drift-diffusion system in 2 dimensional critical space2008

    • Author(s)
      T. Ogawa
    • Organizer
      The third Euro-Japanese work shop on Blow-up
    • Place of Presentation
      Tohoku Univ. Sendai, Japan
    • Related Report
      2008 Annual Research Report
  • [Presentation] Drift-diffusion system in 2 dimensional critical space2008

    • Author(s)
      T. Ogawa
    • Organizer
      The third workshop on Nolininear Partial Differential Equations : Analysis, Computation and Applications
    • Place of Presentation
      Xian, China
    • Related Report
      2008 Annual Research Report
  • [Presentation] Two dimensional drift-diffusion equation in the critical Hardy spaces2007

    • Author(s)
      小川卓克
    • Organizer
      日本数学会 秋季総合分科会 関数方程式分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-24
    • Related Report
      2007 Annual Research Report
  • [Presentation] Multiple existence of positive solutions for semilinear elliptic systems with critical Sobolev exponent2007

    • Author(s)
      石渡通徳
    • Organizer
      日本数学会 秋季総合分科会 関数方程式分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-21
    • Related Report
      2007 Annual Research Report
  • [Book] Mathematical Analysis for the self-similarity and self-organization Surikaiseki Kenkyujyo Kokyuroku Bessatsu 152009

    • Author(s)
      T.Ogawa
    • Total Pages
      205
    • Publisher
      有倫社
    • Related Report
      2009 Annual Research Report
  • [Book] これからの非線形偏微分方程式2007

    • Author(s)
      小薗英雄、三沢正史、小川卓克
    • Total Pages
      305
    • Publisher
      日本評論社
    • Related Report
      2007 Annual Research Report
  • [Book] Asymptotic Analysis and Singularity 1, 2, Advanced Study of Pure Mathemat-ics, 472007

    • Author(s)
      Y. Tsutsumi, E. Yanagida, H. Kozono, K. Tanaka, T. Ogawa (eds)
    • Total Pages
      807
    • Publisher
      Publ. Math Soc. Japan.
    • Related Report
      2007 Annual Research Report

URL: 

Published: 2007-04-01   Modified: 2016-04-21  

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