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Research of the Navier-Stokes equations by using the theory of Fourier analysis and semigroup theory

Research Project

Project/Area Number 11640156
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

YAMAZAKI Masao  Graduate School of Economics, Hitotsubashi University, Professor, 大学院・経済学研究科, 教授 (20174659)

Co-Investigator(Kenkyū-buntansha) YAMADA Hiromichi  Graduate School of Economics, Hitotsubashi University, Professor, 大学院・経済学研究科, 教授 (50134888)
MACHIDA Hajime  Graduate School of Commerce, Hitotsubashi University, Professor, 大学院・商学研究科, 教授 (40090534)
IWASAKI Shiro  Graduate School of Economics, Hitotsubashi University, Professor, 大学院・経済学研究科, 教授 (00001842)
ISHIMURA Naoyuki  Graduate School of Economics, Hitotsubashi University, Professor, 大学院・経済学研究科, 助教授 (80212934)
FUJITA Takahiko  Graduate School of Commerce, Hitotsubashi University, Professor, 大学院・商学研究科, 教授 (50144316)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1999: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsNavier-Stokes equations / Lorentz spaces / Morrey spaces / Weak-* convergence / periodic solutions / almost periodic solutions / exterior domains / stability / *-弱収束 / Besov空間 / Hardy空間
Research Abstract

We studied the Navier-Stokes equation on the whole space or on exterior domains. We have already studied the unique existence and the stability under initieal perturbations of stationary solutions with external forces independent of time. In this research we considered the case where external force depends on the time-variable, and studied the unique existence and the stability of solutions. This research generalizes similar researches on time-periodic solutions and solutions almost periodic in time.
On the whole space we employ the Morrey spaces as the space of solutions, and succeeded in generalizing the results for usual L^p-spaces obtained by Professors Hideo Kozono, Mitsuhiro Nakao and Yasushi Taniuchi. On exterior domains we employ the weak-L^p spaces as the space of solutions, and we succeeded in relaxing the assumptions on the external forces very much, and firstly obtained conditions sufficient for the unique existence of solutions in 3-dimensional exterior domains.
The results for the Morrey spaces can be obtained in a manner similar to that employed in our previous study. On the other hand, in the proof of the results for weak-L^p spaces, it is essential to show that the integral of functions with values in a Banach space converges, where the integral is considered to diverge in general. In order to show this fact, we first consider the family of the Lorentz spaces which generalizes the weak-L^pA spaces, and we improved estimates of L^p-L^q type, which is often employed in previous studies, by using real interpolation, and the we used the duality property between the Lorentz spaces.

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (27 results)

All Other

All Publications (27 results)

  • [Publications] Masao Yamazaki: "The Navier-Stokes equations in the weak-Ln space with time-dependent external force"Mathematische Annalen. 317-4. 635-675 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masao Yamazaki: "Solutions in Morrey spaces of the Navier-Stokes equation with time-dependent external force"Funkcialaj Ekvacioj. 43-3. 419-460 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Hideo Kozono and Masao Yamazaki: "Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in exterior domains"Nonlinear Analysis. 38-8. 959-970 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Naoyuki Ishimura and MasaAki Nakamura: "Nonexistence of monotonic solutions of some third order ODE relevant to the Kuramoto-Sivashinsky equation"Taiwanese Journal of Mathematics. 4-4. 621-625 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takao Hanada,Naoyuki Ishimura and MasaAki Nakamura: "Note on steady solutions of the Eguchi-Oki-Matsumura equation"Proceedings of the Japan Academy, Series A. 76-9. 146-148 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takahiko Fujita,Syoiti Niomiya and Syunji Ito: "Symbolical and geometrical characterizations of Kronecker sequences by using the accelerated Brun's algorithm"Journal of Mathematical Sciences, the University of Tokyo. 7-1. 163-193 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 山崎昌男: "Distributionを初期値とするNavier-Stokes方程式"東京大学数理科学研究科. 71 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masao Yamazaki: "The Navier-Stokes equation in the weak-L^n space with time-dependent external force"Mathematische Annalen. 317-4. 635-675 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masao Yamazaki: "Solutions in Morrey spaces of the Navier-Stokes equation with time-dependent external force"Funkcialaj Ekvacioj. 43-3. 419-460 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Hideo Kozono and Masao Yamazaki: "Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in exterior domains"Nonlinear Analysis. 38-8. 959-970 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Naoyuki Ishimura and MasaAki Nakamura: "Nonexistencs of monotonic solutions of some third order ODE relevant to the Kuramoto-Sivashinsky equation"Taiwanese Journal of Mathematics. 4-4. 621-625 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takao Hanada, Naoyuki Ishimura and MasaAki Nakamura: "Note on steady solutions of the Eguchi-Oki-Matsumura equation"Proceedings of the Japan Academy, Series A. 76-9. 146-148 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takahiko Fujita, Syoiti Ninomiya and Syunji Ito: "Symbolical and geometrical characterizations of Kronecker sequences by using the accelerated Brun's algorithm"Journal of Mathematical Sciences, the University of Tokyo. 7-1. 163-193 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masao Yamazaku: "the Navier-Stokes equation with distributions as initial data"Graduate School of Mathematical Sciences, the University of Tokyo (in Japanese). 71 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masao Yamazaki: "The Navier-Stokes equations in the weak-Ln space with time-dependent external force"Mathematische Annalen. 317-4. 635-675 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Masao Yamazaki: "Solutions in Morrey spaces of the Navier-Stokes equation with time-dependent external force"Funkcialaj Ekvacioj. 43-3. 419-460 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Masao Yamazaki and Zhou XiaoFang: "Semilinear heat equations with distributions in Morrey spaces as initial data"Hokkaido Mathematical Journal. 発表予定.

    • Related Report
      2000 Annual Research Report
  • [Publications] Naoyuki Ishimura and MasaAki Nakamura: "Nonexistence of monotonic solutions of some third order ODE relevant to the Kuramoto-Sivashinsky equation"Taiwanese Journal of Mathematics. 4-4. 621-625 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Takao Hanada,Naoyuki Ishimura and MasaAki Nakamura: "Note on steady solutions of the Eguchi-Oki-Matsumura equation"Proceedings of the Japan Academy, Series A. 76-9. 146-148 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Naoyuki Ishimura: "Remarks on the third-order ODEs relevant to the Kuramoto-Sivashinsky equations"Journal of Differential Equations. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 山崎昌男: "種々の函数空間におけるNavier-Stoke方程式"数学. 51-3. 291-308 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Kozono and M.Yamazaki: "Uniqueness Criterion of weak Solutions to the stationary Navier-Stokes Equations in Exterior Domains"Nonlinear Analysis. 38. 959-970 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Morimoto and N.Ishimura: "Remarks on the blow-up criterion for the 3-D Boussinesq Equations"Mathematical Models and Methods in the Applied Sciences. 9-9. 1323-1332 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Imai,N.Ishimura and T.Ushijima: "Motion of Spirals by crystalline curvature"MZAN Mathematical Modeling and Numerical Analysis. 33-4. 797-806 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Imai,N.Ishimura and T.Ushijima: "A Crystalline Motion of Spiral-Shaped Curves with Symmetry"Journal of Mathematical Analysis and Applications. 240-1. 115-127 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Don C.,Lam CH.and H.Yamada: "Decomposition of the vertex operator algebra V√<2>A3"Journal of Algebra. 222-2. 500-510 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 山崎昌男: "Distributionを初期値とするNavier-Stokes方程式"東京大学大学院数理科学研究科. 65 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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