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The approach of an analytic semigroup for free boundary problems of viscous compressible fluids

Research Project

Project/Area Number 17540156
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionShizuoka University

Principal Investigator

SHIMIZU Senjo  Shizuoka University, Faculty of Engineering, Associate professor, 工学部, 助教授 (50273165)

Co-Investigator(Kenkyū-buntansha) SHIBATA Yoshihiro  Waseda University, School of Science and Engineering, Professor, 理工学術院, 教授 (50114088)
KIKUCHI Koji  Shizuoka University, Faculty of Engineering, Professor, 工学部, 教授 (50195202)
HOSHIGA Akira  Shizuoka University, Faculty of Engineering, Associate professor, 工学部, 助教授 (60261400)
ADACHI Shinji  Shizuoka University, Faculty of Engineering, Associate professor, 工学部, 助教授 (40339685)
NAKAJIMA Toru  Shizuoka University, Faculty of Engineering, Associate professor, 工学部, 助教授 (50362182)
Project Period (FY) 2005 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2006: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsFree boundary problems / Analytic semigroups / Lp-maximal regularity / Stokes equations / Navier-Stokes equations / R-boundedness / Fourier multiplier theorem / 非圧縮性粘性流体 / 最大正則性 / ストークス方程式 / 準線形方程式 / 2相流体 / ノイマン境界条件
Research Abstract

In this research, we consider a free boundary problem for the Navier-Stokes equation which describes the motion of an isolated finite volume of viscous incompressible fluid without taking surface tension into account By using the Lagrange coordinates, free boundary problem is written by the quasi-linear equations on the fixed boundary. Our purpose is to prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data.
To treat quasi-linear equations, first we prove the Lp-Lq maximal regularity of solutions to the linearized problem, which is described by the Neumann problem for the Stokes equation. We consider this problem by analytic semigroup approach. Our main issues is to use R-boundedness and operator valued Fourier multiplier theorem which are recently developed by Weis ('01, Math.Ann.) and Denk-Hieber-Pruss ('03, Mem.AMS).
Based on the Lp-Lq maximal regularity result of the linearized problem, by using the contraction mapping principle, we proved a local in time unique existence theorem for any initial data and external force and a global in time unique existence theorem for some small initial data which are orthogonal to the rigid space in the case where external force vanishes.

Report

(3 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report

Research Products

(32 results)

All 2007 2006 2005 Other

All Journal Article (32 results)

  • [Journal Article] Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      J. Math. Soc. Japan 59

      Pages: 1-34

    • NAID

      10019540463

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a generalized resolvent estimate for the Stokes system with Robin boundary condition2007

    • Author(s)
      Yoshihiro Shibata, Rieko Shimada
    • Journal Title

      J. Math. Soc. Japan 59

      Pages: 469-520

    • NAID

      10019540335

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a free boundary problem for the Navier-Stokes equations2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      Differential Integral Equations 20

      Pages: 241-276

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Free boundary problems for a viscous incompressible fluid2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      RIMS Kokyuroku Bessatsu B1

      Pages: 343-356

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      J. Reine Angew. Math. (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Lp-Lq maximal regularity of the interface problem in a bounded domain2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      RIMS Kokyuroku 1536

      Pages: 58-72

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a free boundary problem for the Navier-Stokes equation2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      Differential Integral Equations 20

      Pages: 241-276

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Lp-Lq maximal regularity of the interface problem in a bounded domain2007

    • Author(s)
      Senjo Shimizu
    • Journal Title

      RIMS Kokyuroku 1536

      Pages: 58-72

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      J. Math. Soc. Japan 59・1

      Pages: 1-34

    • NAID

      10019540463

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On a free boundary problem for the Navier-Stokes equations2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      Differential Integral Equations 20・3

      Pages: 241-276

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the L_p-L_q maximal regularity of the Neumann problem for the Stokes equations in a bounded domain2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      J. Reine Angew. Math. (掲載決定)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain2007

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      MSJ-IRI 2005 Proceedings, Advanced Studies in Pure Mathematics

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Lp-Lq maximal regularity of the interface problem for the Stokes system in a bounded domain2007

    • Author(s)
      Senjo Shimizu
    • Journal Title

      数理解析研究所講究録 流体と気体の数学解析 (掲載受理)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the Stokes equation with Neumann boundary condition2006

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      Banach Center Publications 70

      Pages: 239-250

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Globally in time existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle2006

    • Author(s)
      Toshiaki Hishida, Yoshihiro Shihata
    • Journal Title

      WSEAS Trans. Math. 5

      Pages: 303-307

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the Stokes equation with Neumann boundary condition2006

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      Regularity and other aspects of the Navier-Stokes equations, Banach Center Publications 70

      Pages: 239-250

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] Globally in time existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle2006

    • Author(s)
      Yoshihiro Shibata, Toshiaki Hishida
    • Journal Title

      WSEAS Trans. Math. 5

      Pages: 303-307

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain2006

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      MSJ-IRI 2005 Proceedings, Advanced Studies in Pure Mathematics

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Lp-Lq maximal regularity and viscous incompressible flows with free surface2005

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      Proc. Japan Acad. Ser. A 81

      Pages: 151-155

    • NAID

      40007140161

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the Stokes equation with Neumann boundary condition2005

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      PIMS Kokyuroku 1425

      Pages: 40-53

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the rate of decay of the Oseen semigruop in exterior domains and its application to Navier-Stokes equation2005

    • Author(s)
      Yuko Enomoto, Yoshihiro Shibata
    • Journal Title

      J. Math. Fluid Mech. 7

      Pages: 339-367

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the Stokes and Navier-Stokes equations in a perturbed half-space2005

    • Author(s)
      Takayuki Kubo, Yoshihiro Shihata
    • Journal Title

      Adv. Differential Equations 10

      Pages: 695-720

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the Stokes equation with Neumann boundary condition2005

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      RIMS Kokyuroku 1425

      Pages: 40-53

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation,2005

    • Author(s)
      Yuko Enomoto, Yoshihiro Shibata
    • Journal Title

      J. Math. Fluid Mech. 7

      Pages: 339-367

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the Stokes and Navier-Stokes equations in a perturbed half-space2005

    • Author(s)
      Takayuki Kubo, Yoshihiro Shibata
    • Journal Title

      Adv. Differential Equations 10

      Pages: 695-720

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Lp-Lq maximal regularity and viscous incompressible flows with free surface2005

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      Proc.Japan Acad.Ser.A 81

      Pages: 151-155

    • NAID

      40007140161

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the Stokes equation with Neumann boundary condition2005

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      数理解析研究所講究録 流体と気体の数学解析 1425

      Pages: 40-53

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation2005

    • Author(s)
      Y.Enomoto, Yoshihiro Shibata
    • Journal Title

      J.Math.Fluid Mech. 7・no.3

      Pages: 339-367

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the Stokes and Navier-Stokes equations in a perturbed half-space2005

    • Author(s)
      T.Kubo, Yoshihiro Shibata
    • Journal Title

      Adv.Differential Equations 10・no.6

      Pages: 695-720

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      MSJ-IRI 2005 Proceedings (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the L_p-L_q maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      J. Reine Angew. Math. (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      MSJ-IRI 2005 Proceedings, Advamced Studies in Pure Mathematics (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary

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Published: 2005-03-31   Modified: 2016-04-21  

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