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Large time behavior of solutions to partial differential equations with supercritical nonlinearity

Research Project

Project/Area Number 17K05312
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionTokyo Institute of Technology

Principal Investigator

Miura Hideyuki  東京工業大学, 理学院, 教授 (20431497)

Project Period (FY) 2017-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords非線形偏微分方程式 / 非圧縮性Navier-Stokes方程式 / 弱解 / 正則性 / Navier-Stokes方程式 / 非圧縮性粘性流体 / 偏微分方程式
Outline of Final Research Achievements

We studied the initial-boundary value problem and regularity of the solutions to
the incompressible Navier-Stokes equations. In particular, it was shown that there exist global in time weak solutions for the initial-boundary problem for the Navier-Stokes equations in the three dimensional half space for initial data with infinite energy. As for regularity of the solutions, we showed that if a scaled energy of the initial velocity is sufficiently small in some neighborhood, the weak solution is locally smooth at least for a short time. Furthermore, we obtained new estimates on regions where the weak solutions are smooth for initial data in the weighted spaces of square integrable functions.

Academic Significance and Societal Importance of the Research Achievements

非圧縮性Navier-Stokes方程式は流体力学の基礎方程式としての重要性から古くから多くの研究が行われている.しかし,方程式のもつ非局所性から生じる困難から,エネルギーが無限大となるような特異性をもつ速度場の研究は未解明の部分が多かった.本研究では非局所性を扱う上で鍵となる圧力の評価に関して新しい技術を導入することにより,弱解の時間大域的存在や局所正則性に関する成果を得ることができた.今回用いられた手法は非圧縮性粘性流体の数理解析における今後の研究においても有用になると期待できる.

Report

(8 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (29 results)

All 2024 2023 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (7 results) Journal Article (8 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 8 results,  Open Access: 2 results) Presentation (12 results) (of which Int'l Joint Research: 10 results,  Invited: 12 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Yonsei univeristy(韓国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] University of British Columbia(カナダ)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] University of British Columbia(カナダ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Yonsei university(韓国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] University of British Columbia(カナダ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Yonsei university(韓国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Universite de Bordeaux(フランス)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Regular sets and an epsilon-regularity theorem in terms of initial data for the Navier-Stokes equations2021

    • Author(s)
      Kang Kyungkeun、Miura Hideyuki、Tsai Tai-Peng
    • Journal Title

      Pure and Applied Analysis

      Volume: 3 Issue: 3 Pages: 567-594

    • DOI

      10.2140/paa.2021.3.567

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Local regularity conditions on initial data for local energy solutions of the Navier-Stokes equations2021

    • Author(s)
      Kang Kyungkeun、Miura Hideyuki、Tsai Tai-Peng
    • Journal Title

      Partial Differential Equations and Applications

      Volume: 3 Issue: 1

    • DOI

      10.1007/s42985-021-00127-2

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Estimates for the Navier-Stokes equations in the half-space for nonlocalized data2020

    • Author(s)
      Maekawa Yasunori、Miura Hideyuki、Prange Christophe
    • Journal Title

      Analysis & PDE

      Volume: 13 Issue: 4 Pages: 945-1010

    • DOI

      10.2140/apde.2020.13.945

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Local energy weak solutions for the Navier-Stokes equations in the half-space2019

    • Author(s)
      Maekawa, Y., Miura, H., and Prange, C.
    • Journal Title

      Commun. Math. Phys.

      Volume: 367 Issue: 2 Pages: 517-580

    • DOI

      10.1007/s00220-019-03344-4

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On stability of blow-up solutions of the Burgers vortex type for the Navier-Stokes equations with a linear strain2019

    • Author(s)
      Maekawa Yasunori, Miura Hideyuki, Prange Christophe
    • Journal Title

      Journal of Mathematical Fluid Mechanics

      Volume: 21 Issue: 4

    • DOI

      10.1007/s00021-019-0450-5

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Green tensor of the Stokes system and asymptotics of stationary Navier-Stokes flows in the half space2018

    • Author(s)
      Kang Kyungkeun、Miura Hideyuki、Tsai Tai-Peng
    • Journal Title

      Advances in Mathematics

      Volume: 323 Pages: 326-366

    • DOI

      10.1016/j.aim.2017.10.031

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On Isomorphism for the Space of Solenoidal Vector Fields and Its Application to the Incompressible Flows2018

    • Author(s)
      Maekawa Yasunori、Miura Hideyuki
    • Journal Title

      SIAM Journal on Mathematical Analysis

      Volume: 50 Issue: 1 Pages: 339-353

    • DOI

      10.1137/16m1093537

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] On Uniqueness for the Harmonic Map Heat Flow in Supercritical Dimensions2017

    • Author(s)
      Germain Pierre、Ghoul Tej-Eddine、Miura Hideyuki
    • Journal Title

      Communications on Pure and Applied Mathematics

      Volume: 70 Issue: 12 Pages: 2247-2299

    • DOI

      10.1002/cpa.21716

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Critical norm blow-up for the energy supercritical nonlinear heat equation2024

    • Author(s)
      三浦英之
    • Organizer
      Takamatsu Workshop on Differential Equations and Related Topics
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Blow-up of the critical norm for a supercritical semilinear heat equation2023

    • Author(s)
      三浦英之
    • Organizer
      第19回 非線型の諸問題
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Critical norm blow-up for the energy supercritical nonlinear heat equation2023

    • Author(s)
      三浦英之
    • Organizer
      Geometric Aspects of Partial Differential Equations
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Local regularity conditions on initial data for local energy solutions of the Navier-Stokes equations2023

    • Author(s)
      Hideyuki Miura
    • Organizer
      2023 Winter Workshop on Mathematical Analysis of Fluids
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Estimates of the regular set for Navier-Stokes flows in terms of initial data2020

    • Author(s)
      Miura Hideyuki
    • Organizer
      Vorticity, Rotation and Symmetry (V) - Global Results and Nonlocal Phenomena
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Estimates of the regular set for Navier-Stokes flows in terms of initial data2020

    • Author(s)
      Miura Hideyuki
    • Organizer
      非圧縮性粘性流体の数理解析
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Local energy weak solution for the Navier-Stokes equations and applications2019

    • Author(s)
      Miura Hideyuki
    • Organizer
      RIMS Workshop on Mathematical Analysis in Fluid and Gas Dynamics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Short time regularity of Navier-Stokes flows with locally L3 initial data and applications2019

    • Author(s)
      Miura Hideyuki
    • Organizer
      7th China-Japan Workshop on Mathematical Topics from Fluid Mechanics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Local energy weak solutions for the Navier-Stokes equations in the half-space.2019

    • Author(s)
      Hideyuki Miura
    • Organizer
      Maximal regularity and nonlinear PDE
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Local energy weak solutions for the Navier-Stokes equations in the half-space.2018

    • Author(s)
      Hideyuki Miura
    • Organizer
      International Conference on PDEs from Fluids
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Local energy weak solutions for the Navier-Stokes equations in the half-space2018

    • Author(s)
      Miura Hideyuki
    • Organizer
      第10回 名古屋微分方程式研究集会
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On uniqueness for the supercritical harmonic map heat flow2017

    • Author(s)
      Miura Hideyuki
    • Organizer
      第42回偏微分方程式論札幌シンポジウム
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] East Asian Workshop on PDEs from Kinetics and Continuum Mechanics2023

    • Related Report
      2023 Annual Research Report
  • [Funded Workshop] Mathematical Fluid Mechanics and Related Topics2018

    • Related Report
      2018 Research-status Report

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Published: 2017-04-28   Modified: 2025-01-30  

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