Applied Analysis on the Navier-Stokes Equations and Related Dynamical Systems

• Project/Area Number: 20244006
• 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): YAMADA Michio 京都大学, 数理解析研究所, 教授 (90166736) ; 上田 肇一 京都大学, 数理解析研究所, 助教 (00378960)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥28,210,000 (Direct Cost: ¥21,700,000、Indirect Cost: ¥6,510,000); Fiscal Year 2011: ¥6,890,000 (Direct Cost: ¥5,300,000、Indirect Cost: ¥1,590,000); Fiscal Year 2010: ¥5,720,000 (Direct Cost: ¥4,400,000、Indirect Cost: ¥1,320,000); Fiscal Year 2009: ¥6,890,000 (Direct Cost: ¥5,300,000、Indirect Cost: ¥1,590,000); Fiscal Year 2008: ¥8,710,000 (Direct Cost: ¥6,700,000、Indirect Cost: ¥2,010,000)
• Keywords: ナヴィエ-ストークス方程式 / 解の特異点 / 解の爆発 / ナヴィエーストークス方程式 / 安定性と分岐 / 自己相似解 / 爆発の数値解析 / 数値解析学 / 水面波の分岐 / 水面波の特異性 / ナヴィエ=ストークス方程式 / 逆カスケード / 2次元乱流 / オイラー方程式

Research Abstract

The Navier-Stokes equations and the Euler equations are the master equations of the fluid mechanics. We searched for singularity and pseudo singularity. The Proudman. Johnson equation and reaction-diffusion equation with integral constraint were studied. By numerically computing solutions, we found that a topologically simple solutions exists and they exist only in two-dimensional Navier-Stokes equations and those closely related to it.
Trajectories of fluid particles in a progressive water-waves were computed. What is called the Stokes drift was proved mathematically. Our proof is simpler than any other proof and easily adapted to the numerical computation. Our numerically method enables us to compute not only the gravity waves but also capillary-gravity waves.

Research Products

• [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] A nonlinear boundary-value problem with an integral constraint (2012)
  • Author(s): H.Okamoto
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: (掲載確定)
  • 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
• [Journal Article] Vortices of large scale appearing in the 2D stationary Navier-Stokes equations at large Reynolds numbers (2010)
  • Author(s): Sun-Chul Kim and Hisashi Okamoto
  • Journal Title: Japan J. Indust. Appl. Math Volume: vol.27 Issue: 1 Pages: 47-71
  • DOI: 10.1007/s13160-010-0010-0
  • NAID: 10027314516
  • Peer Reviewed
• [Journal Article] Particle traje ctories around a running cylinder or a sphere (2010)
  • Author(s): Mayumi Shoji, Hisashi Okamoto and Takuya Ooura
  • Journal Title: Fluid Dynamics Research Volume: vol.42 Issue: 2 Pages: 025506-025506
  • DOI: 10.1088/0169-5983/42/2/025506
  • NAID: 10028205798
  • Peer Reviewed
• [Journal Article] Particle trajectories around a running cylinder or a sphere (2010)
  • Author(s): Mayumi Shoji, Hisashi Okamoto, Takuya Ooura
  • Journal Title: Fluid Dynamics Research 42
  • NAID: 10028205798
  • Peer Reviewed
• [Journal Article] Vortices of large scale appearing in the 2D stationary Navier-Stokes equations at large Reynolds numbers (2010)
  • Author(s): Sun-Chul Kim, Hisashi Okamoto
  • Journal Title: Japan J.Indust.Appl.Math. 27 Pages: 47-71
  • NAID: 10027314516
  • Peer Reviewed
• [Journal Article] Particle trajectories around a running cylinder or a sphere (2010)
  • Author(s): M.Shoji, H.Okamoto, T.Ooura
  • Journal Title: Fluid Dynamics Research Volume: 42
  • NAID: 10028205798
  • Peer Reviewed
• [Journal Article] Vortices of large scale appearing in the 2D stationary Navier-Stokes equations at large Reynolds numbers (2010)
  • Author(s): S.-C. Kim, H.Okamoto
  • Journal Title: Japan J.Indus.Appl.Math. Volume: 27 Pages: 47-71
  • NAID: 10027314516
  • Peer Reviewed
• [Journal Article] Particle trajectories around a running cylinder or a sphere, (2010)
  • Author(s): M.Shoji, H.Okamoto, T.Ooura
  • Journal Title: Fluid Dynamics Research 42(掲載確定)(印刷中)
  • NAID: 10028205798
  • Peer Reviewed
• [Journal Article] Vortices of large scale appearing in the 2D stationary Navier-Stokes equations at large Reynolds numbers, (2010)
  • Author(s): Sun-Chul Kim, H.Okamoto
  • Journal Title: Japan Journal Industrial & Applied Mathematics 27(掲載確定)(印刷中)
  • NAID: 10027314516
  • Peer Reviewed
• [Journal Article] Well-posedness of the generalized Proudman-Johnson equation with out viscosity (2009)
  • Author(s): H. Okamoto
  • Journal Title: J. Math. Fluid Mech Volume: vol.11 Issue: 1 Pages: 46-59
  • DOI: 10.1007/s00021-007-0247-9
  • Peer Reviewed
• [Journal Article] Well-posedness of the generalized Proudman-Johnson equation without viscosity (2009)
  • Author(s): H. Okamoto
  • Journal Title: J.Math.Fluid Mech. 11 Pages: 46-59
  • Peer Reviewed
• [Journal Article] Well-posedness of the generalized Proudman-Johnson equation without viscosity, (2009)
  • Author(s): H.Okamoto
  • Journal Title: Journal of Mathematical Fluid Mechanics 11 Pages: 46-59
  • Peer Reviewed
• [Journal Article] Well-posedness of the generalized Proudman-Johnson equation without viscosity (2009)
  • Author(s): H. Okamoto
  • Journal Title: Jouraal of Mathematical Fluid Mechanics 11 Pages: 46-59
  • Peer Reviewed
• [Journal Article] On a gene ralization of the Constantin-Lax-Majda equation (2008)
  • Author(s): H. Okamoto, T. Sakajo, and M. Wunsch
  • Journal Title: Nonlinearity Volume: VOL. 21 Issue: 10 Pages: 2447-2461
  • DOI: 10.1088/0951-7715/21/10/013
  • Peer Reviewed
• [Journal Article] On a generalization of the Constantin-Lax-Majda equation (2008)
  • Author(s): H. Okamoto, T. Sakajo, M. Wunsch
  • Journal Title: Nonlinearity 21
  • NAID: 120000947322
  • Peer Reviewed
• [Journal Article] Blow-up problems in the strained vorticity dynamics and critical exponents
  • Author(s): H. Okamoto
  • Journal Title: Journal of Mathematical Society of Japa
  • NAID: 130003384664
  • Peer Reviewed
• [Journal Article] A nonlinear boundary-vlue problem with an in tegral constraint, to appear in Proceedings of Far-From-Equilibrium Dynamics
  • Author(s): H. Okamoto
  • Journal Title: RIMS Kokyuroku Bessatsu
  • Peer Reviewed
• [Presentation] Traje ctories of fluid particles in a periodic water-wave (2011)
  • Author(s): H. Okamoto
  • Organizer: The 8th East-Asia PdE Con ference
  • Place of Presentation: Pohang, Korea
  • Year and Date: 2011-12-21
• [Presentation] Trajectories of fluid particles in a periodic water-wave (2011)
  • Author(s): H.Okamoto
  • Organizer: The 8th East-Asia PdE Conference
  • Place of Presentation: 浦項工科大学、韓国(招待講演)
  • Year and Date: 2011-12-21
• [Presentation] Finite Difference Methods for Blow-up Problems (2011)
  • Author(s): H. Okamoto
  • Organizer: SIAM Conference on Analysis of Partial Differential Equations
  • Place of Presentation: 米国
  • Year and Date: 2011-11-14
• [Presentation] Finite Difference Methods for Blow-up Problems (2011)
  • Author(s): H.Okamoto
  • Organizer: SIAM Conference on Analysis of Partial Differential Equations
  • Place of Presentation: サンディエゴ・マリオトホテル、米国
  • Year and Date: 2011-11-14
• [Presentation] Vortices of large scale appearing in the 2D stationary Navier-Stokes e quations at large Reynolds numbers (2011)
  • Author(s): H. Okamoto
  • Organizer: The Third Oklahoma PDE Workshop, Oklahoma State University
  • Place of Presentation: 米国
  • Year and Date: 2011-11-12
• [Presentation] Vortices of large scale appearing in the 2D stationary Navier-Stokes equations at large Reynolds numbers (2011)
  • Author(s): H.Okamoto
  • Organizer: The Third Oklahoma PDE Workshop
  • Place of Presentation: オクラホマ州立大学、米国(招待講演)
  • Year and Date: 2011-11-12
• [Presentation] The mathematicians grapple with the Navier-Stokes equations (2011)
  • Author(s): H.Okamoto
  • Organizer: 明治大学先端数理科学研究科開設記念シンポジウム記念講演会「現象数理学の将来への展望」
  • Place of Presentation: 明治大学駿河台キャンパス(招待講演)
  • Year and Date: 2011-10-05
• [Presentation] A Survey on Blow-up Problems around the Navier-Stokes Equation (2011)
  • Author(s): H. Okamoto
  • Organizer: International Congress of Industrial and Applied Mathematics
  • Place of Presentation: Canada
  • Year and Date: 2011-07-22
• [Presentation] A Survey on Blow-up Problems around the Navier-Stokes Equation (2011)
  • Author(s): H.Okamoto
  • Organizer: International Congress of Industrial and Applied Mathematics
  • Place of Presentation: Vancouver Convention Center, Canada
  • Year and Date: 2011-07-22
• [Presentation] Vortices of large scale appearing in the 2D stationary Navier-Stokes equations at large Reynolds numbers (2011)
  • Author(s): 岡本久
  • Organizer: 早稲田大学流体数学セミナー
  • Place of Presentation: 早稲田大学理工学部
  • Year and Date: 2011-07-11
• [Presentation] A curious bifurcation diagram arising in a scalar reaction-diffusion equation with an integral constraint (2011)
  • Author(s): H.Okamoto
  • Organizer: The 7th East Asia SIAM Conference
  • Place of Presentation: 早稲田大学北九州キャンパス
  • Year and Date: 2011-06-28
• [Presentation] The vorticity dynamics of in compressible fluid in linear strain : blow-up and a critical exponent (2011)
  • Author(s): H. Okamoto
  • Organizer: 4th Euro-Japanese Workshop on Blow-up Leiden
  • Place of Presentation: オランダ
  • Year and Date: 2011-06-09
• [Presentation] ストークス漂流について (2011)
  • Author(s): 岡本久
  • Organizer: 慶応大学非線形解析セミナー
  • Place of Presentation: 慶応大学
  • Year and Date: 2011-06-03
• [Presentation] Blow-up solutions of the Navier-Stokes equations with in finite energy (2011)
  • Author(s): H. Okamoto
  • Organizer: Recent Develo pments in Nonlinear Partial Differential Equations
  • Place of Presentation: 香港中文大学香港
  • Year and Date: 2011-05-16
• [Presentation] Blow-up solutions of the Navier-Stokes equations with infinite energy (2011)
  • Author(s): H.Okamoto
  • Organizer: Recent Developments in Nonlinear Partial Differential Equations : Part II
  • Place of Presentation: 香港中文大学、香港(招待講演)
  • Year and Date: 2011-05-16
• [Presentation] Blow-up problems in the strained vorticity dynamics and critical Exponents (2011)
  • Author(s): H. Okamoto
  • Organizer: 2011 Spring Western Section Meeting University of Nevada
  • Place of Presentation: Las Vegas, NV米国
  • Year and Date: 2011-04-30
• [Presentation] Blow-up problems in the strained vorticity dynamics and critical exponents (2011)
  • Author(s): H.Okamoto
  • Organizer: 2011 Spring Western Section Meeting
  • Place of Presentation: ネヴァダ大学、米国(招待講演)
  • Year and Date: 2011-04-30
• [Presentation] Pattern for mations at high Reynolds numbers : Navier-Stokes, Kolmogorov (2011)
  • Author(s): H. Okamoto
  • Organizer: 京都大学数理解析研究所研究集会
  • Place of Presentation: Far-From-Equilibrium Dynamics
  • Year and Date: 2011-01-04
• [Presentation] Pattern formations at high Reynolds numbers : Navier-Stokes, Kolmogorov, and… (2011)
  • Author(s): H.Okamoto
  • Organizer: Far-From-Equilibrium Dynamics
  • Place of Presentation: 京都大学(招待講演)
  • Year and Date: 2011-01-04
• [Presentation] Navier-Stokes and Kolmogorov (2011)
  • Author(s): H. Okamoto
  • Organizer: International Conference on Far-from-Equilibrium Dynamics
  • Place of Presentation: Kyoto Japan
• [Presentation] Stokes driftについて (2010)
  • Author(s): H.Okamoto
  • Organizer: 非線形波動現象の多様性と普遍性
  • Place of Presentation: 京都大学
  • Year and Date: 2010-10-13
• [Presentation] The vorticity dynamics of incompressible fluid in linear strain : blow-up and a critical exponent (2010)
  • Author(s): H.Okamato
  • Organizer: 大韓民国仁荷大学数学教室談話会
  • Place of Presentation: 仁荷大学(大韓民国)
  • Year and Date: 2010-09-15
• [Presentation] The vorticity dynamics of incompressible fluid in linear strain (2010)
  • Author(s): H. Okamoto
  • Organizer: The 4th Euro-Japanese Workshop on Blow-up
  • Place of Presentation: Leiden University,オランダ王国
  • Year and Date: 2010-09-06
• [Presentation] The vorticity dynamics of incompressible fluid in linear strain (2010)
  • Author(s): H.Okamoto
  • Organizer: 4th Euro-Japanese Workshop on Blow-up
  • Place of Presentation: ライデン大学ローレンツセンター(オランダ)(招待講演)
  • Year and Date: 2010-09-06
• [Presentation] On the Stokes drift (2010)
  • Author(s): H. Okamoto
  • Organizer: 第3回日中韓数値数学国際会議
  • Place of Presentation: 大韓民国江陵市
  • Year and Date: 2010-08-19
• [Presentation] On the Stokes drift (2010)
  • Author(s): H.Okamato
  • Organizer: 第3回日中韓数値数学国際会議
  • Place of Presentation: 江陵大学(大韓民国)
  • Year and Date: 2010-08-19
• [Presentation] Steady-States of the Navier-Stokes equations which may be related to 2D turbulence (2010)
  • Author(s): H.Okamoto
  • Organizer: Mathematics in Science and Technology
  • Place of Presentation: イスラム文化センター、ニューデリー(インド)(招待講演)
  • Year and Date: 2010-08-15
• [Presentation] The generalized Proudman-Johnson equation at large Reynolds numbers (2010)
  • Author(s): H.Okamoto
  • Organizer: The 6th East Asia SIAM Conference
  • Place of Presentation: クアラルンプール大学(マレーシア)
  • Year and Date: 2010-06-22
• [Presentation] On a generalized Constantin-Lax-Majda equation (2010)
  • Author(s): H.Okamoto
  • Organizer: The 10th international workshop on differential equations-in memory of the late Prof. Jeongseon Baek
  • Place of Presentation: Chonnam National University, Korea.
  • Year and Date: 2010-03-19
• [Presentation] 非線形発展方程式の解の爆発問題に対する差分近似 (2010)
  • Author(s): 岡本久
  • Organizer: 数値シミュレーションの理論と実践
  • Place of Presentation: 九州大学西新プラザ
  • Year and Date: 2010-02-16
• [Presentation] 一般化されたConstantin-Lax-Majda方程式について. (2010)
  • Author(s): 岡本久
  • Organizer: 北陸応用数理研究会2010
  • Place of Presentation: 金泥大学サテライトプラザ
  • Year and Date: 2010-02-12
• [Presentation] The Navier-Stokes equations and related paradoxes (2010)
  • Author(s): H.Okamoto
  • Organizer: First Friendship Meeting in Mathematics between Fudan and Kyoto Universities
  • Place of Presentation: 京都大学百周年時計台記念館国際交流ホール
  • Year and Date: 2010-01-11
• [Presentation] Fluid mechanical paradoxes and their an alytical reso lutions (2009)
  • Author(s): H. Okamoto
  • Organizer: PDE and Functional Analysis Research Center
  • Place of Presentation: Korea
  • Year and Date: 2009-11-25
• [Presentation] Fluid mechanical paradoxes and their analytical resolutions. (2009)
  • Author(s): H.Okamoto
  • Organizer: PDE and Functional Analysis Research Center Kick-off seminar
  • Place of Presentation: Seoul University, Korea
  • Year and Date: 2009-11-25
• [Presentation] Particle trajectories around a running cylinder or a sphere (2009)
  • Author(s): H. Okamoto
  • Organizer: Taiwan-Japan Joint Workshop on Numerical Analysis and Scientific Computation
  • Place of Presentation: Taiwan University,台湾
  • Year and Date: 2009-11-07
• [Presentation] Particle trajectories around a running cylinder or a sphere. (2009)
  • Author(s): H.Okamoto
  • Organizer: Taiwan-Japan Joint Workshop on Numerical Analysis and Scientific Computation
  • Place of Presentation: Taiwan University, Taiwan
  • Year and Date: 2009-11-07
• [Presentation] Particle trajectories around a running cylinder or a sphere. (2009)
  • Author(s): H.Okamoto
  • Organizer: 日本流体力学会年会
  • Place of Presentation: 東洋大学白山キャンパス
  • Year and Date: 2009-09-04
• [Presentation] Pattern formations in the Kolmogorov flow. (2009)
  • Author(s): H.Okamoto
  • Organizer: 第34回偏微分方程式論札幌シンポジウム
  • Place of Presentation: 北海道大学理学部
  • Year and Date: 2009-08-25
• [Presentation] Bifurcation of Kolmogorov flows and large vortices (2009)
  • Author(s): H. Okamoto
  • Organizer: The 5th East Asia SIAM Conference
  • Place of Presentation: University of Brunei, Brunei
  • Year and Date: 2009-06-09
• [Presentation] Bifurcation of Kolmogorov flows and large vortices. (2009)
  • Author(s): H.Okamoto
  • Organizer: The 5th East Asia SIAM Conference
  • Place of Presentation: University of Brunei, Brunei
  • Year and Date: 2009-06-09
• [Presentation] Dedicate to Prof Andrew Majda's 60th birthday (2009)
  • Author(s): H. Okamoto
  • Organizer: The International Conference on Contemporary Applied Mathe matics
  • Place of Presentation: Shanghai, China
• [Presentation] Stationary vortices of large scale at large Reynolds numbers (2009)
  • Author(s): H. Okamoto
  • Organizer: The International Conference on Contemporary Applied Mathematics
  • Place of Presentation: Fudan University, Shanghai,中華人民共和国
• [Presentation] The in viscid limit of steady-states of the Navier-Stokesequations (2008)
  • Author(s): H. Okamoto
  • Organizer: Analytical and Computational Fluid Dynamics 2008
  • Place of Presentation: Chung-Ang University, Korea
• [Presentation] On mathe matical issues on discre tization of the blow-up problems (2008)
  • Author(s): H. Okamoto
  • Organizer: Workshop on Foundations of Numerical PDEs City University of Hong Kong
• [Presentation] Uniqueness of Crapper's pure capillary wave of permanent form (2008)
  • Author(s): H. Okamoto
  • Organizer: Nonlinear Partial Differential Equations : Analysis, Computation and Applications
  • Place of Presentation: Northwest University,西安、中華人民共和国
• [Book] ストークス方程式の数理 (2009)
  • Author(s): ナヴィエ
  • Publisher: 東京大学出版会
• [Book] ナヴィエ-ストークス方程式の数理 (2009)
  • Author(s): 岡本久
  • Total Pages: 365
  • Publisher: 東京大学出版会
• [Remarks]
  • URL: http://www.kurims.kyoto-u.ac.jp/~okamoto/
• [Remarks]
  • URL: http://www.kurims.kyoto-u.ac.jp/~okamoto/