Mathematical research on mechanical problems of incompressible fluid

• Project/Area Number: 18H01137
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Gakushuin University
• Principal Investigator: Okamoto Hisashi 学習院大学, 理学部, 教授 (40143359)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000); Fiscal Year 2021: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2020: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2019: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
• Keywords: Navier-Stokes equations / 非線形偏微分方程式 / 流体力学 / 水面波 / Prandtl-Batchelor theory / 精度保証計算 / 水面波の解析 / ナヴィエ・ストークス方程式 / Navier-Stokes方程式 / 数値計算法 / 数理流体力学 / 数値解析学 / Navier-Stokes 方程式

Outline of Final Research Achievements

The Kolmogorov problems for the two-dimensional Navier-Stokes equations was studied and a kind of universal patterns for large vortices were confirmed by the numerical experiments. Water-waves on two vortical layers were computed numerically, and many highly nontrivial solutions of many stagnation points were discovered.New method of computing the blow-up time for nonlinear parabolic equations were proposed.

Academic Significance and Societal Importance of the Research Achievements

特異摂動の問題に極めて不思議な解が存在することを数値実験で確かめた。それを保証する数学的な理論も部分的に提出された。しかし、完全な解決には程遠い。これらは次の世代への問題と言うことができる。特にプラントル・バチェラー理論の一般化には大きな魅力を感じる。

Research Products

• [Int'l Joint Research] Chung-Ang University(韓国)
• [Journal Article] A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution (2022)
  • Author(s): Mean Sovanna、Unami Koichi、Okamoto Hisashi、Fujihara Masayuki
  • Journal Title: Applied Mathematics and Computation Volume: 415 Pages: 126730-126730
  • DOI: 10.1016/j.amc.2021.126730
  • Peer Reviewed / Open Access
• [Journal Article] Rigorous numerics for nonlinear heat equations in the complex plane of time (2022)
  • Author(s): Takayasu Akitoshi、Lessard Jean-Philippe、Jaquette Jonathan、Okamoto Hisashi
  • Journal Title: Numerische Mathematik Volume: 151 Issue: 3 Pages: 693-750
  • DOI: 10.1007/s00211-022-01291-2
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Prandtl-Batchelor Theory for Kolmogorov Flows (2020)
  • Author(s): Kim Sun-Chul、Okamoto Hisashi
  • Journal Title: Journal of the Physical Society of Japan Volume: 89 Issue: 11 Pages: 114401-114401
  • DOI: 10.7566/jpsj.89.114401
  • NAID: 40022406698
  • Peer Reviewed / Open Access
• [Journal Article] Finite difference schemes for an axisymmetric nonlinear heat equation with blow-up (2020)
  • Author(s): Cho Chien-Hong、Okamoto Hisashi
  • Journal Title: ETNA - Electronic Transactions on Numerical Analysis Volume: 52 Pages: 391-415
  • DOI: 10.1553/etna_vol52s391
  • Peer Reviewed / Open Access
• [Journal Article] Existence proof of unimodal solutions of the Proudman?Johnson equation via interval analysis (2019)
  • Author(s): Miyaji T, Okamoto H
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Volume: 36 Issue: 1 Pages: 287-298
  • DOI: 10.1007/s13160-018-00339-x
  • NAID: 210000173644
  • Peer Reviewed
• [Journal Article] Weal periodic solutions and numerical case studies of the Fornberg-Whitham equation (2019)
  • Author(s): G. Hoermann & H. Okamoto
  • Journal Title: Disc. Conti. Dynam. Sys. A, Volume: 39 Issue: 8 Pages: 4455-4469
  • DOI: 10.3934/dcds.2019182
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Models and Special Solutions of the Navier-Stokes Equations (2018)
  • Author(s): Hisashi Okamoto
  • Journal Title: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids Volume: 2 Pages: 729-780
  • DOI: 10.1007/978-3-319-13344-7_14
  • ISBN: 9783319133430, 9783319133447
  • Peer Reviewed
• [Journal Article] Unimodal solutions of the generalized Constantin?Lax?Majda equation with viscosity (2018)
  • Author(s): Kim S-C, Miyaji T, Okamoto Hi
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Volume: 35 Issue: 3 Pages: 1065-1083
  • DOI: 10.1007/s13160-018-0320-1
  • NAID: 210000172939
  • Peer Reviewed / Int'l Joint Research
• [Presentation] An application of the Prantdl-Batchelor theory to the Kolmogorov flows (2019)
  • Author(s): Hisashi Okamoto
  • Organizer: East Asia SIAM Conference
  • Int'l Joint Research
• [Presentation] An application of the Prandtl-Batchelor theory to the Kolmogorov flows (2019)
  • Author(s): Hisashi Okamoto
  • Organizer: Int. Conference on Modeling, Computations, Theoretical Analysis on Fluid Dynamics and Related Problems
  • Int'l Joint Research / Invited
• [Presentation] Numerical experiments for weak solutions of the Fornberg-Whitham equation (2019)
  • Author(s): Hisashi Okamoto
  • Organizer: Kyoto univ.--Chung-Ang Univ. Conference on Nonlinear PDEs
  • Int'l Joint Research / Invited
• [Presentation] Numerical experiments for weak solutions of the Fornberg-Whitham equation (2019)
  • Author(s): Hisashi Okamoto
  • Organizer: Modeling, applied analysis and computation of fluids, SUSTech, China
  • Int'l Joint Research / Invited
• [Presentation] Mathematical issues on high-Reynolds number flows (2019)
  • Author(s): Hisashi Okamoto
  • Organizer: Int. Congress on Industrial and Applied Mathematics, Valencia, Spain
  • Int'l Joint Research / Invited
• [Presentation] Numerical experiments for weak solutions of the Fornberg-Whitham equation (2019)
  • Author(s): Hisashi Okamoto
  • Organizer: American Mathematical Society Regional conference, University of Hawaii
  • Invited
• [Presentation] Existence proof of solutions of fluid mechanics problems via interval analysis (2018)
  • Author(s): Hisashi Okamoto
  • Organizer: Conference
  • Invited
• [Presentation] Numerical computations of water waves with piecewise constant vorticity (2018)
  • Author(s): Hisashi Okamoto
  • Organizer: Taiwan SIAM Annual Conference
  • Invited
• [Presentation] Two-dimensional Large Eddies : a Mathematical Challenge from Fluid Mechanics (2018)
  • Author(s): Hisashi Okamoto
  • Organizer: IBS Workshop for Nonlinear PDE, Korea
  • Invited
• [Presentation] Numerical computation of water waves on two vortical layers (2018)
  • Author(s): Hisashi Okamoto
  • Organizer: A3 Workshop on mathematic of Biology, Fluid Dunamics, and Material Sciences Gangneung, Korea
  • Invited
• [Presentation] Unimodal solutions of the equations of 2D incompressible fluid motion (2018)
  • Author(s): Hisashi Okamoto
  • Organizer: CAMIS International Conference on Partial Differential Equations Theories, Numerics and Applications
  • Invited
• [Presentation] Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation (2018)
  • Author(s): Hisashi Okamoto
  • Organizer: International Conference in Mathematics and Applications
  • Invited
• [Remarks]
  • URL: https://www-cc.gakushuin.ac.jp/~20170001/index.htm