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2005 Fiscal Year Final Research Report Summary

Mathematical analysis on the structure of solutions for the fundamental systems of equations in continuum mechanics

Research Project

Project/Area Number 15340050
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKEIO UNIVERSITY

Principal Investigator

TANI Atusi  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (90118969)

Co-Investigator(Kenkyū-buntansha) KIKUCHI Norio  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (80090041)
SIMOMOURA Shun  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (00154328)
NODERA Takashi  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (50156212)
ISHIKAWA Shiro  Keio Univ., Faculty of Sci.and Tech., Associate Professor, 理工学部, 助教授 (10051913)
TAKAYAMA Masahiro  Keio Univ., Faculty of Sci.and Tech., Assistant, 理工学部, 助手 (90338252)
Project Period (FY) 2003 – 2005
KeywordsNavier-Stokes equations / Euler equation / Free boundary problems / infinite sector / Contact line / Functions of bounded variation / nonlinear acoustics
Research Abstract

Among the fundamental equations in continuum mechanics we have obtained the following results.
1.Since it is well known that the evolution problems of isentropic Euler equation admit shock waves even if the initial data are smooth, we usually try to find the solution belonging to the function spaces of bounded variations. In order to guarantee the uniqueness of the solution it is convenient to construct such a solution as a limit of the solution to the approximate equation of parabolic type. We succeeded to construct the temporally global solution in the class of functions of bounded variations to this approximate equation. Up to the present time we have had a scenario due to Nishida and Smoller to construct such a solution. However, their scenario is valid only if the density is bounded. We firstly succeeded to prove its boundedness, so that in real sense their scenario works.
2.Among the two-dimensional evolution free boundary problems for incompressible viscous fluid we study the case … More where the free boundary and the boundary of the container has a contact line. As a series of our study on the solvability of Navier-Stokes equations in a container with slip boundary conditions, here we investigated the two problems :
(1)Stokes equations in infinite sector,
(2)Navier-Stokes equations in a domain with piecewise smooth boundary.
Then we proved their solvability in the weighted Sobolev spaces.
3.For the one-dimensional model equations of a self-gravitating viscous radiative and reactive gas we found the unique global in time solution belonging to Hoelder spaces. For this problem we used the Stefan-Boltzmann relation.
4.In a two-dimensional infinite elastic or visco-elastic strip with a semi-infinite crack we studied the solvability to the stationary problem and determined the propagation of the crack. Moreover, we proved the weak solvability of its evolution problem.
Now the following results are preparing : (1)To construct the solution around the Gerstner's trochoidal wave and 3D domain for incompressible inviscid flow (2)Nonlinear problems in nonlinear acoustics. Less

  • Research Products

    (12 results)

All 2006 Other

All Journal Article (12 results)

  • [Journal Article] Finite difference schemes for a certain nonlinear parabolic system2006

    • Author(s)
      H.Morioka, A.Tani
    • Journal Title

      Math.Models Meth.Appl.Sci 16・5(未定)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On some boundary value problem for the Stokes equations in an infinite sector2006

    • Author(s)
      S.Itoh, N.Tanaka, A.Tani
    • Journal Title

      Analysis Appl. 4 ・3(未定)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Finite difference schemes for a certain nonlinear parabolic system2006

    • Author(s)
      H.Morioka, A.Tani
    • Journal Title

      Math.Models Meth.Appl.Sci. 16-5(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On some boundary value problem for the Stokes equations in an infinite sector2006

    • Author(s)
      S.Itoh, N.Tanaka, A.Tani
    • Journal Title

      Analysis Appl. 4-3(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas2006

    • Author(s)
      M.Umehara, A.Tani
    • Journal Title

      Kokyuroku (RIMS Kyoto University) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Shape derivative of energy on crack in an infinite elastic strip with a semi-infinite cracks2006

    • Author(s)
      H.Itou, A.Tani
    • Journal Title

      Tokyo J. 29-1(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Global solution to the one-dimensional model equations for viscous liquid stars in radiative and reactive processes2006

    • Author(s)
      M.Umehara, A.Tani
    • Journal Title

      Kokyuroku (RIMS Kyoto University) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Steady Navier-Stokes equations in a domain with piecewise smooth boundary

    • Author(s)
      S.Itoh, N.Tanaka, A.Tani
    • Journal Title

      Computers and Mathematics with Applications to appear

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas

    • Author(s)
      M.Umehara, A.Tani
    • Journal Title

      京都大学教理解析研究所講究録 to appear

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Shape derivative of energy on crack in an infinite elastic strip with a semi-infinite cracks

    • Author(s)
      H.Itou, A.Tani
    • Journal Title

      Tokyo J. to appear

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 燃焼過程を伴う一次元粘性流体星モデル方程式の時間大域解

    • Author(s)
      梅原 守道, 谷温之
    • Journal Title

      京都大学数理解析研究所講究録 to appear

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Steady Navier-Stokes equations in a domain with piecewise smooth boundary

    • Author(s)
      S.Itoh, N.Tanaka, A.Tani
    • Journal Title

      Computers and Mathematics with Applications (to appear)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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