Applied analysis for nonlinear problems

• Project/Area Number: 15K21369
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis; Foundations of mathematics/Applied mathematics
• Research Institution: Keio University
• Principal Investigator: SOGA Kohei 慶應義塾大学, 理工学部(矢上), 専任講師 (80620559)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: Hamilton力学系 / Hamilton-Jacobi方程式 / 粘性解 / 弱KAM理論 / Navier-Stokes方程式 / 最大正則性 / Leray-Hopfの弱解 / 差分法 / discount 近似 / 有限差分法 / 収束証明 / 誤差評価 / discount近似 / 二相自由境界問題 / 適切性 / modeling

Outline of Final Research Achievements

(1) We studied Hamiltonian dynamics and Hamilton-Jacobi equations in terms of weak KAM theory. We formulated a version of weak KAM theory for discounted Hamilton-Jacobi equations and applied it to quantitative analysis of the vanishing discount limit. We also prepared necessary mathematical concepts in order to construct a version of weak KAM theory for discretized Hamilton-Jacobi equations.

(2) We did mathematical analysis on two phase flows of compressible fluids in terms of the maximal regularity. We also presented a numerical method of incompressible fluids and proved its convergence.

Academic Significance and Societal Importance of the Research Achievements

(1) discount近似および差分近似されたHamilton-Jacobi方程式に対して弱KAM理論の枠組みを与えることで、同研究分野の理論的・数値解析的方法のさらなる発展に貢献した。

(2) 自由境界を持つNavier-Stokes方程式の厳密な数学解析行うと同時に、流体力学における数値計算の数学的正当性を保証する理論を構築することで、同研究分野の理論的・数値解析的方法のさらなる発展に貢献した。

Research Products

• [Int'l Joint Research] ダルムシュタット工科大学(ドイツ)
• [Journal Article] Weak KAM theory for discount Hamilton-Jacobi equations and its application (2018)
  • Author(s): H. Mitake, K. Soga
  • Journal Title: Calc. Var. Partial Differential Equations Volume: accepted Issue: 3
  • DOI: 10.1007/s00526-018-1359-1
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface (2016)
  • Author(s): Takayuki Kubo, Yoshihiro Shibata, Kohei Soga
  • Journal Title: Discrete and Continuous Dynamical Systems - Series A (DCDS-A) Volume: Volume 36, Issue 7 Issue: 7 Pages: 3741-3774
  • DOI: 10.3934/dcds.2016.36.3741
  • NAID: 120007135602
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] On convergence of Chorin’s projection method to a Leray-Hopf weak solution -Bounded Lipschitz domain case- (2018)
  • Author(s): Kohei Soga
  • Organizer: Conference on Mathematical Fluid Dynamics Bad Boll
  • Int'l Joint Research / Invited
• [Presentation] A Finite Difference Method in Hamilton-Jacobi Equations and Weak KAM Theory (2018)
  • Author(s): Kohei Soga
  • Organizer: 12th AIMS Conference in Taipei
  • Int'l Joint Research / Invited
• [Presentation] On convergence of Chorin's projection method to a Leray-Hopf weak solution (2018)
  • Author(s): 曽我幸平
  • Organizer: JSPS-DFG Japanese-German Graduate Externship "The 15th Japanese-German International Workshop on Mathematical Fluid Dynamics"
  • Int'l Joint Research / Invited
• [Presentation] ハミルトン・ヤコビ方程式のディスカウント近似に対する選択問題 : 収束率 (2017)
  • Author(s): 曽我幸平 (発表者), 三竹大寿
  • Organizer: 日本数学会2017年度年会
  • Place of Presentation: 首都大学東京 （東京都 八王子市）
  • Year and Date: 2017-03-25
• [Presentation] エントロピー解・粘性解の選択問題 (2017)
  • Author(s): 曽我幸平
  • Organizer: 「数学と現象：Mathematics and Phenomena in Miyazaki 2017
  • Invited
• [Presentation] 弱KAM理論の応用 1 - HJ方程式の放物型近似・差分近似・discount近似と対応する力学系 (2016)
  • Author(s): 曽我幸平
  • Organizer: RIMS研究集会: 力学系とその関連分野の連携探索
  • Place of Presentation: 京都大学 （京都府 京都市）
  • Year and Date: 2016-06-08
  • Invited
• [Presentation] 古典KAM理論・弱KAM理論入門 (2016)
  • Author(s): 曽我幸平
  • Organizer: RIMS研究集会: 力学系とその関連分野の連携探索
  • Place of Presentation: 京都大学 （京都府 京都市）
  • Year and Date: 2016-06-07
  • Invited
• [Presentation] Local well-posedness and global well-posedness of two-phase flows: compressible-compressible case (2015)
  • Author(s): Kohei Soga
  • Organizer: RIMS 研究集会「非圧縮性粘性流体の数理解析」
  • Place of Presentation: 京都大学数理解析研究所（京都府 京都市）
  • Year and Date: 2015-11-16