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On regularity and uniqueness of solutions to partial differential equations in Fluid Mechanics and Harmonic Analysis

Research Project

Project/Area Number 16K05228
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionShinshu University

Principal Investigator

Taniuchi Yasushi  信州大学, 学術研究院理学系, 教授 (80332675)

Project Period (FY) 2016-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords関数方程式論 / 偏微分方程式 / 流体力学 / Navier-Stokes方程式 / 熱対流方程式 / 非圧縮性流体 / 非圧縮性粘性流体 / 関数方程式 / 流体
Outline of Final Research Achievements

We founded the weakest norm that satisfies the Brezis-Gallouet-Wainger type inequality, under some conditions. As an application of the Brezis-Gallouet-Wainger type inequality, we gave Beale-Kato-Majda type blow-up criteria of smooth solutions to the 3-D Navier-Stokes equations in unbounded domains. For example, we proved that, if [0,T) is the maximal interval of existence of a smooth solution u to the Navier-Stokes equations, then
int_(0,T) || rot u(s)||_{bmo}ds= infty.
Moreover, we improved this blow-up criterion by using a space of Morrey type.

Academic Significance and Societal Importance of the Research Achievements

水や油などの縮まない流体の運動を記述するNavier-Stokes方程式の解の性質に関する研究をおこなった。 この方程式は数学のみならす、物理学、工学、気象学等の様々な自然科学の分野で利用される極めて重要な方程式である。 また、同方程式の滑らかな解の大域存在は、数学の7つの未解決問題(いわいるミレニアム問題)に選ばれており、同方程式の研究は数学分野においても重要視されていることがわかる。

Report

(6 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (12 results)

All 2021 2020 2019 2018 2017 2016

All Journal Article (3 results) (of which Peer Reviewed: 2 results) Presentation (8 results) (of which Int'l Joint Research: 4 results,  Invited: 8 results) Funded Workshop (1 results)

  • [Journal Article] An alternative proof of logarithmically improved Beale?Kato?Majda type extension criteria for smooth solutions to the Navier?Stokes equations2018

    • Author(s)
      Nakao Kohei、Taniuchi Yasushi
    • Journal Title

      Nonlinear Analysis

      Volume: 176 Pages: 48-55

    • DOI

      10.1016/j.na.2018.05.018

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Brezis-Gallouet-Wainger Type Inequalities and Blow-Up Criteria for Navier-Stokes Equations in Unbounded Domains2018

    • Author(s)
      Nakao Kohei、Taniuchi Yasushi
    • Journal Title

      Communications in Mathematical Physics

      Volume: 359 Issue: 3 Pages: 951-973

    • DOI

      10.1007/s00220-017-3061-0

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in bounded domains2016

    • Author(s)
      K. Nakao, Y. Taniuchi
    • Journal Title

      数理研考究録

      Volume: 2009 Pages: 44-52

    • Related Report
      2016 Research-status Report
  • [Presentation] On uniqueness of mild solutions on the whole time axis to the Boussinessq equations in unbounded domains2021

    • Author(s)
      Yasushi Taniuchi
    • Organizer
      非圧縮性粘性流体の数理解析
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Some logarithmic inequalities and the Navier-Stokes equations2020

    • Author(s)
      Taniuchi, Y.
    • Organizer
      第21回北東数学解析研究会
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Some logarithmic inequalities and the Navier-Stokes equations2020

    • Author(s)
      Taniuchi, Y.
    • Organizer
      松山解析セミナー2020
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Some logarithmic inequalities and the Navier-Stokes equations2019

    • Author(s)
      Taniuchi, Y.
    • Organizer
      Maximal regularity and nonlinear PDE
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Brezis-Gallouet-Wainger type inequality and its application to Navier-Stokes equations in unbounded domains2017

    • Author(s)
      Y. Taniuchi
    • Organizer
      VORTICITY, ROTATION AND SYMMETRY (IV) - Complex Fluids and the Issue of Regularity
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Brezis-Gallouet-Wainger type inequality and its application to the Navier-Stokes equations2017

    • Author(s)
      Y. Taniuchi
    • Organizer
      保存則をもつ偏微分方程式に対する解の正則性,特異性および漸近挙動の研究
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Brezis-Gallouet-Wainger type inequality and its application to Navier-Stokes equations in unbounded domains2016

    • Author(s)
      Y. Taniuchi
    • Organizer
      第六回弘前非線形方程式研究会
    • Place of Presentation
      弘前大学創立50周年記念会会館 岩木ホール
    • Year and Date
      2016-12-23
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in unbounded domains2016

    • Author(s)
      Y. Taniuchi
    • Organizer
      New trends in Partial Differential Equations
    • Place of Presentation
      the Centro De Giorgi in Pisa、イタリア
    • Year and Date
      2016-10-06
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Mathematical Fluid Mechanics and Related Topics2018

    • Related Report
      2018 Research-status Report

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Published: 2016-04-21   Modified: 2022-01-27  

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