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2004 Fiscal Year Final Research Report Summary

Applications of the dynamical systems theory and the singularity theory to mathematical fluid mechanics

Research Project

Project/Area Number 14204007
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyoto University

Principal Investigator

OKAMOTO Hisashi  Kyoto University, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (40143359)

Co-Investigator(Kenkyū-buntansha) OHKITANI Koji  Kyoto University, RIMS, Associate Professor, 数理解析研究所, 助教授 (70211787)
UEDA Keiichi  Kyoto University, RIMS, Assistant Professor, 数理解析研究所, 助手 (00378960)
NAGAYAMA Masaharu  Kanazawa Univ., Dept.of Comput.Sci., Associate Professor, 大学院・自然科学研究科, 助教授 (20314289)
KIMURA Yoshifumi  Nagoya Univ., Dept.Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70169944)
SAKAJOU Takashi  Hokkaido Univ., Dept.Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (10303603)
Project Period (FY) 2002 – 2004
KeywordsNavier-Stokes equations / Euler equations / dynamical systems / singularity / vortex sheet / trajectories of fluid particles / reaction-diffusion system / pulse solutions
Research Abstract

The Navier-Stokes equations are the fundamental partial differential equations of fluid mechanics. The regularity problem of the solutions of the equations is particularly renowned but the equations also accompany other equally important issues. The present research aims at better understanding of the equations and related reaction-diffusion equations. We have achieved substantial progresses in (1)discovery of solutions of new kind, particularly nearly singular solutions possessing internal layers, (2)bifurcation analysis for progressive water waves and a development of new computational methods for water waves. S.-C.Kim and Okamoto considered rhombic periodic Navier-Stokes flows, and discovered a curious stability-exchange at large Reynolds numbers. A.Craik and Okamoto studied a three dimensional dynamical system stemming from the Navier-Stokes equations. Asymptotic behavior of its solutions was classified : a dichotomy was discovered and the reason was explained by the existence of a periodic solution. X.Chen and Okamoto proved that the Proudman-Johnson equation did not admit a blow up under the Dirichlet boundary condition, which resolved an open problem for ten years.
Nagayama and Okamoto studied certain axisymmetric self-similar solutions of the Navier-Stokes equations, and proved that they possessed internal-layers for large Reynolds numbers. K.-I.Nakamura, H.Yagisita and Okamoto rigorously proved that self-similar solutions of the Navier-Stokes equations blew up in the framework of the Burgers vortex. Okamoto proved a uniqueness theorem for Crapper's waves arising in the progressive water waves.

  • Research Products

    (9 results)

All 2004 2003 2002

All Journal Article (8 results) Book (1 results)

  • [Journal Article] Motion of a vortex sheet on a sphere with pole vortices2004

    • Author(s)
      T.Sakajo
    • Journal Title

      Physics of Fluids 16

      Pages: 717-727

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] particle transport by a vortex soliton2004

    • Author(s)
      Kimura, Y., Koikari, S.
    • Journal Title

      J.Fluid Mechanics 510

      Pages: 201-218

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Phase-dependent output of scattering process for traveling breathers2004

    • Author(s)
      T.Teramoto, K.-I.Ueda, Y.Nishiura
    • Journal Title

      Physical Review E 69

      Pages: 056224-1-056224-8

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Numerical computation of water and solitary waves by the double exponential transform2003

    • Author(s)
      K.Kobayashi, N.Okamoto, J.Zhu
    • Journal Title

      Journal of Computational and Applied Mathematics 152

      Pages: 229-241

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Numerical computation of water and solitary waves by the double exponential transform2003

    • Author(s)
      K.Kobayashi, H.Okamoto, J.Zhu
    • Journal Title

      Journal of Computational and Applied Mathematics 152

      Pages: 229-241

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the interior layer appearing in the similarity solutions of the Navier-Stokes equations2002

    • Author(s)
      M.Nagayama, H.Okamoto
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 19

      Pages: 277-300

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A three-dimensional autonomous system with unbounded 'bending' solutions2002

    • Author(s)
      A.D.D.Craik, H.Okamoto
    • Journal Title

      Physica D 164

      Pages: 168-186

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A three-dimensional autonomous system with unbounded ‘bending' solutions2002

    • Author(s)
      A.D.D.Craik, H.Okamoto
    • Journal Title

      Physica D 164

      Pages: 168-186

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 現象の数理2003

    • Author(s)
      岡本 久
    • Total Pages
      230
    • Publisher
      放送大学教育振興会
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2006-07-11  

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