Applied analysis of equations of incompressible fluid motions

• Project/Area Number: 24244007
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kyoto University
• Principal Investigator: Okamoto Hisashi 京都大学, 数理解析研究所, 教授 (40143359)
• Co-Investigator(Kenkyū-buntansha): 山田 道夫 京都大学, 数理解析研究所, 教授 (90166736)
• Co-Investigator(Renkei-kenkyūsha): SHOJI Mayumi 日本女子大学, 理学部, 教授 (10090523)
• Research Collaborator: SAKAJO Takashi ; NAGAYAM Masaharu ; MIYAJI Tomoyuki ; KIM Sun-Chul ; Craik Alexander D. D.
• Project Period (FY): 2012-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥27,300,000 (Direct Cost: ¥21,000,000、Indirect Cost: ¥6,300,000); Fiscal Year 2016: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000); Fiscal Year 2015: ¥6,110,000 (Direct Cost: ¥4,700,000、Indirect Cost: ¥1,410,000); Fiscal Year 2014: ¥5,590,000 (Direct Cost: ¥4,300,000、Indirect Cost: ¥1,290,000); Fiscal Year 2013: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2012: ¥5,850,000 (Direct Cost: ¥4,500,000、Indirect Cost: ¥1,350,000)
• Keywords: 数理流体力学 / 2次元流れ / 高レイノルズ数 / 分岐 / 流体 / 解析学 / ナヴィエ－ストークス方程式 / 解の分岐 / 水面波 / 流体力学 / 応用数学 / 特異摂動

Outline of Final Research Achievements

The Navier-Stokes equations, which are the master equations of the fluid motion, are considered. In the following four problems, I obtained significant results. [a] discovery of unimodal solutions, [b] study on well-posedness of various model equations for fluid motion,[c] numerical method for computing blow-up solutions, [d] numerical computation of water-waves.

Research Products

• [Journal Article] Unimodal patterns appearing in the two-dimensional Navier-Stokes flows under general forcing at large Reynolds numbers, (2017)
  • Author(s): S.-C. Kim, H. Okamoto, T. Miyaji
  • Journal Title: European J. Mechanics, B Fluid Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Models and Special Solutions of the Navier-Stokes Equations, (2017)
  • Author(s): Hisashi Okamoto
  • Journal Title: Handbook on of Mathematical Analysis in Mechanics of Viscous Fluid Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A blow-up problem for a nonlinear heat equation in the complex plane of time (2016)
  • Author(s): C.-H. Cho, H. Okamoto, & M. Shoji
  • Journal Title: Japan J. Indust. Appl. Math. Volume: 33 Issue: 1 Pages: 145-166
  • DOI: 10.1007/s13160-015-0203-7
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] 巨大渦の安定性―２次元非圧縮高レイノルズ数の流れの中で (2016)
  • Author(s): 岡本 久
  • Journal Title: 日本物理学会誌 Volume: 71 Pages: 526-532
  • NAID: 130005282504
  • Peer Reviewed
• [Journal Article] A four-leaf chaotic attractor of a three-dimensional dynamical system (2015)
  • Author(s): Tomoyuki Miyaji, Hisashi Okamoto, Alex D.D. Craik
  • Journal Title: Int. J. Bifurcation and Chaos Volume: 25 Issue: 02 Pages: 1530003-1530003
  • DOI: 10.1142/s0218127415300037
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Three-dimensional forced-damped dynamical systems with rich dynamics: bifurcations, chaos and unbounded solutions, (2015)
  • Author(s): T. Miyaji, H. Okamoto, and A. D. D. Craik
  • Journal Title: Physica D Volume: 311-312 Pages: 25-36
  • DOI: 10.1016/j.physd.2015.09.001
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Unimodal patterns appearing in the Kolmogorov flows at large Raynolds numbers (2015)
  • Author(s): S.-C. Kim and H. Okamoto
  • Journal Title: Nonlinearity Volume: 28 Issue: 9 Pages: 3219-3242
  • DOI: 10.1088/0951-7715/28/9/3219
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Steady-States and Traveling wave solutions of the generalized Constantin-Lax-Majda equation (2014)
  • Author(s): Hisashi Okamoto, T. Sakajo, M. Wunsch
  • Journal Title: Discrete and Continuous Dynamical Systems, B Volume: 34 Issue: 8 Pages: 3155-3170
  • DOI: 10.3934/dcds.2014.34.3155
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] The generalized Proudman-Johnson equation and its singular perturbation problems (2014)
  • Author(s): Sun -Chul Kim, Hisashi Okamoto
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Volume: 31 Issue: 3 Pages: 541-573
  • DOI: 10.1007/s13160-014-0146-4
  • NAID: 210000172941
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A computer-assisted proof of existence of a periodic solution (2014)
  • Author(s): Tomoyuki Miyaji, Hisashi Okamoto
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. Volume: 90 Issue: 10 Pages: 139-144
  • DOI: 10.3792/pjaa.90.139
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] The generalized Proudman--Johnson equation and its singular perturbation (2014)
  • Author(s): S.-C. Kim and H. Okamoto
  • Journal Title: Japan J. Indust. Appl. Math. Volume: 31
  • Peer Reviewed
• [Journal Article] Blow-up problems in the strained vorticity dynamics and critical exponents (2013)
  • Author(s): H. Okamoto
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 65 Issue: 4 Pages: 1079-1099
  • DOI: 10.2969/jmsj/06541079
  • NAID: 130003384664
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] ２次元非圧縮粘性流における、安定で大きな渦の存在について (2013)
  • Author(s): 岡本 久、Kim Sun-Chul
  • Journal Title: ながれ Volume: 32 Pages: 417-419
• [Journal Article] Trajectories of fluid particles in a periodic water-wave (2012)
  • Author(s): H.Okamoto, M.Shoji
  • Journal Title: Phil.Trans.R.Soc.Lond., A Volume: 370 Issue: 1964 Pages: 1661-1676
  • DOI: 10.1098/rsta.2011.0447
  • NAID: 110008460050
  • Peer Reviewed
• [Journal Article] The ge neralized Proudman-Johnson e quation at large Reynolds numbers. (2011)
  • Author(s): S.-C. Kim and H. Okamoto
  • Journal Title: Online Volume: (online) Issue: 2 Pages: 379-403
  • DOI: 10.1093/imamat/hxr054
  • Peer Reviewed
• [Presentation] Some applications of very accurate numerical methods in fluid mechanics (2016)
  • Author(s): Hisashi Okamoto
  • Organizer: Modeling and Simulation of Hierarchical and Heterogeneous Flow Systems with Applications to Materials Science III
  • Place of Presentation: 東北大学 知の館
  • Year and Date: 2016-11-14
  • Int'l Joint Research
• [Presentation] Generalized Constantin-Lax-Majda equation with viscosity (2016)
  • Author(s): Hisashi Okamoto
  • Organizer: Annual Conference of SIAM East Asia Section
  • Place of Presentation: Macau, China
  • Year and Date: 2016-06-21
  • Int'l Joint Research
• [Presentation] Some mathematical issues about special solutions of the Navier-Stokes equations (2016)
  • Author(s): Hisashi Okamoto
  • Organizer: The Navier-Stokes Equations and Related Topics, Nagoya University
  • Place of Presentation: 名古屋大学
  • Year and Date: 2016-03-08
  • Int'l Joint Research / Invited
• [Presentation] 数理流体力学に現れる困難について (2015)
  • Author(s): 岡本 久
  • Organizer: 日本数学会総合講演
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
  • Invited
• [Presentation] Unimodal patterns appearing in the 2D Navier-Stokes equations at large Reynolds numbers, I (2015)
  • Author(s): Hisashi Okamoto
  • Organizer: International Congress for Industrial and Applied Mathematics
  • Place of Presentation: 北京、中華人民共和国
  • Year and Date: 2015-08-10
  • Int'l Joint Research
• [Presentation] A nonlinear heat equation in the complex plane and an associated ill-posed problem (2015)
  • Author(s): Hisashi Okamoto
  • Organizer: 台湾応用数理学会年会
  • Place of Presentation: 高雄、台湾
  • Year and Date: 2015-05-30
  • Int'l Joint Research / Invited
• [Presentation] An application of interval arithmetic to certain dynamical systems arising in fluid mechanics (2015)
  • Author(s): Hisashi Okamoto
  • Organizer: The 9th International Conference on Computational Physics
  • Place of Presentation: National University of Singapore
  • Year and Date: 2015-01-07 – 2015-01-11
  • Invited
• [Presentation] Unimodal patterns appearing in the Kolmogorov flows at large Reynolds numbers (2014)
  • Author(s): Hisashi Okamoto
  • Organizer: Colloquium
  • Place of Presentation: School of Mathematixs, University of Minnesota, USA
  • Year and Date: 2014-11-20
  • Invited
• [Presentation] Topologically simple patterns appearing in the 2D Navier-Stokes equations at large Reynolds numbers (2014)
  • Author(s): Hisashi Okamoto
  • Organizer: The 8th CREST-SBM International Conference International Conference on Mathematical Fluid Dynamics
  • Place of Presentation: 早稲田大学
  • Year and Date: 2014-11-11 – 2014-11-14
  • Invited
• [Presentation] Trajectories of fluid particles around spheres and in a water wave (2014)
  • Author(s): Hisashi Okamoto
  • Organizer: Theory of Water Waves
  • Place of Presentation: Isaac Newton Institute, Cambridge, UK
  • Year and Date: 2014-07-22
• [Presentation] Unimodal Patterns appearing in the 2D Navier-Stokes equations at large Reynolds numbers (2014)
  • Author(s): Hisashi Okamoto
  • Organizer: Conference of East Asia Section of SIAM
  • Place of Presentation: Ambassador City Hotel Jomtien, Pattaya, Chonburi, Thailand
  • Year and Date: 2014-06-23 – 2014-06-25
  • Invited
• [Presentation] A blow-up problem on the nonlinear heat equation in the complex t-plane (2013)
  • Author(s): H. Okamoto
  • Organizer: The 9th East Asia SIAM Conference
  • Place of Presentation: Bandung Institute of Technology (Indonesia)
• [Presentation] 2次元非圧縮粘性流における安定で大きな渦の存在について (2013)
  • Author(s): 岡本 久
  • Organizer: 日本流体力学会年会
  • Place of Presentation: 東京農工大
• [Presentation] Fluid mechanical singular perturbation problems and its numerical challenges (2013)
  • Author(s): H. Okamoto
  • Organizer: 北東アジアシンポジウム（Method and Modeling for High performance Scientific Computing）
  • Place of Presentation: 望江賓館 (中国 成都市）
  • Invited
• [Presentation] Kolmogorov's Dynamical Systems viewpoint on 2D Navier-Stokes flows. (2013)
  • Author(s): H. Okamoto
  • Organizer: International Conference on PDEs and Dynamical Systems
  • Place of Presentation: 河南理工大学（中国 焦作市）
  • Invited
• [Presentation] Nonlinear heat equations, blow-ups, and their finite difference approximations (2013)
  • Author(s): H. Okamoto
  • Organizer: The first Vietnam International Applied Mathematics Conference
  • Place of Presentation: サイゴン大学（ホーチミン市）
  • Invited
• [Presentation] One-dimensional model equations for incompressible fluid motion (2013)
  • Author(s): H. Okamoto
  • Organizer: UK-Japan Winter School: Nonlinear Analysis
  • Place of Presentation: Royal Academy of Engineering, London、（イギリス）
  • Invited
• [Presentation] A finite difference scheme for an axisymmetric nonlinear heat equation with blow-up (2013)
  • Author(s): H. Okamoto
  • Organizer: CAU-Kyoto University Joint Workshop on Nonlinear PDEs
  • Place of Presentation: Seoul、Chung-Ang University, （大韓民国）
  • Invited
• [Presentation] A Pattern Formation in2D Navier-Stokes Equations at very high Reynolds numbers (2012)
  • Author(s): H. Okamoto
  • Organizer: The 8th East Asia SIAM Conference
  • Place of Presentation: 台湾大学、台北、（台湾）
• [Presentation] 2次元非圧縮流におけるある種の普遍法則 (2012)
  • Author(s): 岡本 久
  • Organizer: 非線形現象の数理と数値解析2012
  • Place of Presentation: 富山大学
  • Invited
• [Presentation] Pattern formation of the Kolmogorv flows at high Reynolds numbers (2012)
  • Author(s): H. Okamoto
  • Organizer: Nonlinear Partial Differential Equations, Dynamical Systems and Their Applications
  • Place of Presentation: 京都大学
  • Invited
• [Book] 関数とは何か (2014)
  • Author(s): 岡本 久、長岡 亮介
  • Total Pages: 325
  • Publisher: 近代科学社