Mathematical analysis of dispersion and anisotropy in rotating fluids

• Project/Area Number: 25887005
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Mathematical analysis
• Research Institution: Tohoku University
• Principal Investigator: TAKADA RYO 東北大学, 理学(系)研究科(研究院), 助教 (50713236)
• Project Period (FY): 2013-08-30 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 偏微分方程式 / 流体力学 / 非圧縮性流体 / 回転流体 / Navier-Stokes 方程式 / Euler 方程式 / Coriolis 力 / 分散型評価 / 温度成層 / Boussinesq 方程式 / 非圧縮性 Euler 方程式 / 非圧縮性 Navier-Stokes 方程式 / Strichartz 評価

Outline of Final Research Achievements

This research project aims to analyze nonlinear partial differential equations describing the motion of incompressible rotating fluids mathematically. We established dispersive and space-time estimates for the linear propagator generated by the Coriolis force. As an application, we proved the long time existence of solutions to the initial value problem for the incompressible rotating Euler equations. It is expected that our results and techniques obtained in this research can be applied to the study of the rotating shallow water equations and the primitive equations.

Research Products

• [Journal Article] Long time existence of classical solutions for the 3D incompressible rotating Euler equations (2015)
  • Author(s): Ryo Takada
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 掲載確定
  • NAID: 130006887147
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Dispersive estimates for the Navier-Stokes equations in the rotational framework (2014)
  • Author(s): Youngwoo Koh, Sanghyuk Lee, Ryo Takada
  • Journal Title: Advances in Differential Equations Volume: 19 Pages: 857-878
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type (2014)
  • Author(s): Tsukasa Iwabuchi, Ryo Takada
  • Journal Title: Journal of Functional Analysis Volume: 267 Issue: 5 Pages: 1321-1337
  • DOI: 10.1016/j.jfa.2014.05.022
  • Peer Reviewed
• [Journal Article] Strichartz estimates for the Euler equations in the rotational framework (2014)
  • Author(s): Youngwoo Koh, Sanghyuk Lee, Ryo Takada,
  • Journal Title: Journal of Differential Equations Volume: 256 Issue: 2 Pages: 707-744
  • DOI: 10.1016/j.jde.2013.09.017
  • Peer Reviewed
• [Journal Article] Existence of periodic solutions and their asymptotic stability to the Navier-Stokes equations with the Coriolis force (2014)
  • Author(s): Kozono,H., Mashiko,Y., Takada,R.
  • Journal Title: J. Evolution Equations Volume: 14 Issue: 3 Pages: 565-601
  • DOI: 10.1007/s00028-014-0228-4
  • Peer Reviewed
• [Journal Article] Stability of time periodic solutions for the rotating Navier-Stokes equations (2014)
  • Author(s): Tsukasa Iwabuchi, Alex Mahalov, Ryo Takada
  • Journal Title: Recent Developments of Mathematical Fluid Mechanics, Series: Advances in Mathematical Fluid Mechanics Volume: 掲載確定
  • Peer Reviewed
• [Journal Article] Dispersive effect of the Coriolis force and the local wellposedness for the Navier-Stokes equations in the rotational framework (2014)
  • Author(s): Tsukasa Iwabuchi, Ryo Takada
  • Journal Title: Funkcialaj Ekvacioj Volume: 掲載確定
  • NAID: 130005020755
  • Peer Reviewed
• [Journal Article] Global solutions for the Navier-Stokes equations in the rotational framework (2013)
  • Author(s): Tsukasa Iwabuchi, Ryo Takada
  • Journal Title: Mathematische Annalen Volume: (掲載確定) Issue: 2 Pages: 727-741
  • DOI: 10.1007/s00208-013-0923-4
  • Peer Reviewed
• [Journal Article] Dispersive effect of the Coriolis force for the Navier-Stokes equations in the rotational framework (2013)
  • Author(s): Tsukasa Iwabuchi, Ryo Takada
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B42
  • Peer Reviewed
• [Journal Article] Global solutions for the rotating Navier-Stokes equations (2013)
  • Author(s): Tsukasa Iwabuchi, Ryo Takada
  • Journal Title: RIMS Kokyuroku Volume: 1830
• [Presentation] Long time existence of classical solutions to the 3D rotating Euler equations (2015)
  • Author(s): 高田 了
  • Organizer: NLPDE セミナー
  • Place of Presentation: 京都大学（京都市）
  • Year and Date: 2015-01-16
  • Invited
• [Presentation] 回転流体に現れる分散性の数学解析 (2014)
  • Author(s): 高田 了
  • Organizer: 第77回分野横断セミナー
  • Place of Presentation: 東北大学流体科学研究所（仙台市）
  • Year and Date: 2014-12-24
  • Invited
• [Presentation] Long time existence of classical solutions to the 3D rotating Euler equations (2014)
  • Author(s): 高田 了
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 京都大学（京都市）
  • Year and Date: 2014-12-20 – 2014-12-21
  • Invited
• [Presentation] Remarks on the Strichartz estimates for the rotating incompressible fluids (2014)
  • Author(s): Ryo Takada
  • Organizer: Autumn School and Workshop on Mathematical Fluid Dynamics
  • Place of Presentation: The Protestant Academy Bad Boll (Bad Boll, Germany)
  • Year and Date: 2014-10-27 – 2014-10-30
  • Invited
• [Presentation] Remarks on the Strichartz estimates for rotating incompressible fluids (2014)
  • Author(s): 高田 了
  • Organizer: 第8 回実解析と函数解析による微分方程式研究集会
  • Place of Presentation: 鳴子ホテル（大崎市）
  • Year and Date: 2014-10-03 – 2014-10-05
• [Presentation] Long time existence of classical solutions to the 3D rotating Euler equations (2014)
  • Author(s): 高田 了
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学（熊本市）
  • Year and Date: 2014-09-06
  • Invited
• [Presentation] Long time solvability for the 3D rotating Euler equations (2014)
  • Author(s): Ryo Takada
  • Organizer: The 10th AIMS conference on Dynamical Systems, Differential Equations and Applications. SS83 Fluid flows in unbounded domains
  • Place of Presentation: The Instituto de Ciencias Matematicas (Madrid, Spain)
  • Year and Date: 2014-07-07 – 2014-07-11
  • Invited
• [Presentation] Long time solvability of the 3D incompressible Euler equations with high-speed rotation (2014)
  • Author(s): Ryo Takada
  • Organizer: Harmonic Analysis and Nonlinear Partial Differential Equations
  • Place of Presentation: 京都大学数理解析研究所（京都市）
  • Year and Date: 2014-06-30 – 2014-07-02
  • Invited
• [Presentation] Long time solvability for the 3D rotating Euler equations (2014)
  • Author(s): Ryo Takada
  • Organizer: VORTICITY, ROTATION AND SYMMETRY (III) - Approaching Limiting Cases of Fluid Flows
  • Place of Presentation: the Centre International de Rencontres Mathematiques (Marseille, France)
  • Year and Date: 2014-05-05 – 2014-05-09
  • Invited
• [Presentation] Strichartz estimates for incompressible rotating fluids (2014)
  • Author(s): 高田 了
  • Organizer: 調和解析駒場セミナー
  • Place of Presentation: 東京大学（目黒区）
  • Year and Date: 2014-04-19
  • Invited
• [Presentation] Long time existence for the 3D incompressible Euler equations with high-speed rotation (2014)
  • Author(s): 高田 了
  • Organizer: 若手による流体力学の基礎方程式研究集会
  • Place of Presentation: 名古屋大学（名古屋市）
  • Invited
• [Presentation] The 3D rotating Euler equations with the critical Sobolev regularity (2014)
  • Author(s): Ryo Takada
  • Organizer: The 15th Northeastern Symposium on Mathematical Analysis
  • Place of Presentation: 北海道大学（札幌市）
  • Invited
• [Presentation] 回転流体に現れる分散性の数学解析 (2014)
  • Author(s): 高田 了
  • Organizer: 流体と幾何学
  • Place of Presentation: 早稲田大学（新宿区）
  • Invited
• [Presentation] Strichartz estimates for incompressible rotating fluids (2014)
  • Author(s): 高田 了
  • Organizer: 調和解析学駒場セミナー
  • Place of Presentation: 東京大学（目黒区）
  • Invited
• [Presentation] Long time solvability for the 3D rotating Euler equations (2014)
  • Author(s): Ryo Takada
  • Organizer: VORTICITY, ROTATION AND SYMMETRY (III) - Approaching Limiting Cases of Fluid Flows
  • Place of Presentation: the Centre International de Rencontres Mathematiques（ルミニー、フランス）
  • Invited
• [Presentation] Long time solvability for the 3D rotating Euler equations (2014)
  • Author(s): Ryo Takada
  • Organizer: The 10th AIMS conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: the Instituto de Ciencias Matematicas（マドリッド、スペイン）
  • Invited
• [Presentation] Long time existence for the 3D incompressible Euler equations with high-speed rotation (2013)
  • Author(s): Ryo Takada
  • Organizer: Mathematical Analysis of Viscous Incompressible Fluid
  • Place of Presentation: 京都大学数理解析研究所（京都市）
  • Invited