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非線形放物型方程式のアトラクターの研究

Research Project

Project/Area Number 18654031
Research Category

Grant-in-Aid for Exploratory Research

Allocation TypeSingle-year Grants
Research Field Basic analysis
Research InstitutionWaseda University

Principal Investigator

大谷 光春  Waseda University, 理工学術院, 教授 (30119656)

Co-Investigator(Kenkyū-buntansha) 北田 韶彦  早稲田大学, 理工学術院, 教授 (10123118)
田中 和永  早稲田大学, 理工学術院, 教授 (20188288)
Project Period (FY) 2006 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywords無限次元力学系 / 非線形現象 / 放物型偏微分方程式
Research Abstract

研究目的にかかげた目標に関する次の幾つかの興味ある成果が得られた。
(1)我々の先行研究によって、p-Laplacianを主要項に持つ準線形放物型方程式
u_t=△_p u+uに対する初期値境界値問題に対して、全ての解軌道を引き付ける「大域アトラクター」が、L^2で構成され、さらにそれが無限次元を持つ事実が知られていたが、これはかなり特殊な状況であり、非線形楕円型方程式に関するLyusternik-Scnirelman理論からも、その無限次元性は導出できるという難点があった。
uをαu-b(x)|u|^q uとしても、大域アトラクターの存在とその無限次元性が導かれる
ことが示された。これは、より一般的な非線形項f(x, u)に対しても、同様な結果が成立することを示唆する、重要な発見である。
(2)多孔質媒質中を流れる流体(溶媒)の速度及び温度と流体中の溶質の濃度の振舞いを記述する、2または3次元有界領域におけるBrin kman-Forchheimer方程式の時間大域解の存在と一意性が、H^1に属する初期値に対して、示された。
これによって、この方程式に対する、大域アトラクターの構成の出発点がクリアーされたことになる。
また、よく知られているように、3次元空間におけるナビエ・ストークス方程式の一意的時間大域解の存在問題が未解決大問題である事実と比較すると、非常に興味深い知見を与えている。

Report

(3 results)
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (24 results)

All 2009 2008 2007 2006 Other

All Journal Article (11 results) (of which Peer Reviewed: 6 results) Presentation (12 results) Book (1 results)

  • [Journal Article] On the Navier-Stokes equation with siip-like boundary condition2009

    • Author(s)
      T. Koyama, M. Otani, K. Hashizume
    • Journal Title

      Electoric Journal of Differential Equations to appear

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Parablic equations with singularity on the boundary2008

    • Author(s)
      C. P. Aroeo, J. M. L. Esoaner IV, M. Otani, P. W. Sy
    • Journal Title

      Advances in Mathematicak Sciences and Applications 18

      Pages: 29-42

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Note on a property specific to the tent map2008

    • Author(s)
      Kitada, Ogasawara, Eda
    • Journal Title

      Chaos Solitons & Fractals 35

      Pages: 104-105

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Infinite dimensional attractors for evolution equations with p-Laplacian and their Kolmogorov entropy2007

    • Author(s)
      M.Efendiev and M. Otani
    • Journal Title

      Differential and Integral Equations 25

      Pages: 1201-1209

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Exponential attractors for a quasilinear parabolic equation2007

    • Author(s)
      K.Matsuura and M.Otani
    • Journal Title

      Dynamical Systems, Differential Equations and Applications supplement 2007

      Pages: 713-720

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a dendrite generated by a zero-dimensional weak self-similar set2007

    • Author(s)
      Kitada, Ogasawara, Yamamoto
    • Journal Title

      Chaos Solitons & Fractals 34

      Pages: 1732-1735

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] High frequency solutions for singularly perturbed 1D nonlinear Schrodinger equation2006

    • Author(s)
      P.Felmer, S.Marinez, K.Tanaka
    • Journal Title

      Arch. Rat. Mech. Anal. 182

      Pages: 333-366

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the number of positive solutions of singularly perturbed 1D Nonlinear Schrodinger equations2006

    • Author(s)
      P.Felmer, S.Marinez, K.Tanaka
    • Journal Title

      J. Euro Math. Soc. 8

      Pages: 253-268

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Existence of nontrivial solutions or some elliptic equations with supercritical nonlinearity in exterior domains

    • Author(s)
      Satoshi Hashimoto, Mitsuharu Otani
    • Journal Title

      Discrete and Continuous Dvnamical System (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On a dendrite generated by a zero-dimensional weak self-similar set

    • Author(s)
      A.Kitada, Y.Ogasawara, T.Yamamoto
    • Journal Title

      Chaos Solitons & Fractals (in print)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Note on a property specific to the tent map

    • Author(s)
      A.Kitada, Y.Ogasawara, K.Eda
    • Journal Title

      Chaos Solitons & Fractals (in print)

    • Related Report
      2006 Annual Research Report
  • [Presentation] 非線形境界条件下での楕円型の解の存在について2009

    • Author(s)
      J. Harada, M. Otani
    • Organizer
      函数方程式分科会/日本数学会
    • Place of Presentation
      東京大学
    • Year and Date
      2009-03-26
    • Related Report
      2008 Annual Research Report
  • [Presentation] Infinite dimensionality of attractors for evolution equations governed by p-Laplacian2009

    • Author(s)
      M. Otani
    • Organizer
      Evolution Equations and Dynamical Systems, Conference on the occasion of the 60th birthday of Alain Haraux
    • Place of Presentation
      Hammamet, Tunisia
    • Year and Date
      2009-03-24
    • Related Report
      2008 Annual Research Report
  • [Presentation] Some parabolic equations with nonlinear boundary conditions2009

    • Author(s)
      M. Otani
    • Organizer
      International Conference “Schrodinger Equations and Related Topics" on the occasion of Professor Kenji Yajima's sixtieth birthday
    • Place of Presentation
      Univ. of Tokyo
    • Year and Date
      2009-01-08
    • Related Report
      2008 Annual Research Report
  • [Presentation] Smoothing effect and uniqueness for some doubly nonlinear parabolic equations2008

    • Author(s)
      M. Otani
    • Organizer
      International Conference of Mathematical Analysison Chemotaxis and Semicon duotor simulation
    • Place of Presentation
      RIMS Kyoto Univ.
    • Year and Date
      2008-09-17
    • Related Report
      2008 Annual Research Report
  • [Presentation] Long-time behavior of solutions of some doubly nonlinear parabolic equations2008

    • Author(s)
      M. Otani
    • Organizer
      WCNA 2008 in Orlando
    • Place of Presentation
      Orlando, Florida USA
    • Year and Date
      2008-07-04
    • Related Report
      2008 Annual Research Report
  • [Presentation] Semilinear elliptic equations with nonlinear boundary conditions2008

    • Author(s)
      J. Harada, M. Otani
    • Organizer
      WCNA 2008 in Orlando
    • Place of Presentation
      Orlando Florida USA
    • Year and Date
      2008-07-04
    • Related Report
      2008 Annual Research Report
  • [Presentation] H^2-solutions for some elliptic equations with nonlinear boundary conditions2008

    • Author(s)
      J. Harada, M. Otani
    • Organizer
      7th AIMS International Conference on Dynamical Systems, Differential Equations and Applications 2008
    • Place of Presentation
      Univ. of Texas (UTA)
    • Year and Date
      2008-05-19
    • Related Report
      2008 Annual Research Report
  • [Presentation] 弱い自己相似集合から逐次形成されるグラフの諸性質2008

    • Author(s)
      北田 韶彦
    • Organizer
      日本数学会
    • Place of Presentation
      近畿大学
    • Related Report
      2007 Annual Research Report
  • [Presentation] L∞-エネルギー法について-その基礎と応用-2007

    • Author(s)
      大谷 光春
    • Organizer
      日本数学会 企画特別講演
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Related Report
      2007 Annual Research Report
  • [Presentation] 境界条件に非線形性を持つ楕円型の解の存在について2007

    • Author(s)
      原田 潤一、大谷 光春
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-03-21
    • Related Report
      2007 Annual Research Report
  • [Presentation] 弱い自己相似集合からのグラフの逐次形成2007

    • Author(s)
      北田 韶彦
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Related Report
      2007 Annual Research Report
  • [Presentation] A positive solution of a scalar field equation:-Δu=g(x,u)in R^-N2007

    • Author(s)
      K.Tanaka
    • Organizer
      Chile-Japan PDE symposium
    • Place of Presentation
      University of Chile
    • Related Report
      2007 Annual Research Report
  • [Book] Variational Problems and Applications2007

    • Author(s)
      Chjan C.Lim, M.Otani and J.Shi
    • Total Pages
      467
    • Publisher
      Discrete and Continuous Dynamical Systems-Series A
    • Related Report
      2007 Annual Research Report

URL: 

Published: 2006-04-01   Modified: 2016-04-21  

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