Construction of mathematical theory to investigate the macro structure and the mesostructure of the fluid motion

2016 Fiscal Year Final Research Report

• Project/Area Number: 24224004
• Research Category: Grant-in-Aid for Scientific Research (S)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Waseda University
• Principal Investigator: Shibata Yoshihiro 早稲田大学, 理工学術院, 教授 (50114088)
• Co-Investigator(Kenkyū-buntansha): 田端 正久 早稲田大学, 理工学術院, 教授 (30093272) ; 吉村 浩明 早稲田大学, 理工学術院, 教授 (40247234) ; 舟木 直久 東京大学, 数理(科)学研究科(研究院), 教授 (60112174) ; 小澤 徹 早稲田大学, 理工学術院, 教授 (70204196)
• Co-Investigator(Renkei-kenkyūsha): YAMAZAKI Masao 早稲田大学, 理工学術院, 教授 (20174659) ; HISHIDA Toshiaki 名古屋大学, 多元数理科学研究科, 教授 (60257243) ; SHIMIZU Senjo 京都大学, 人間・環境学研究科, 教授 (50273165) ; SUZUKI Yukihito 早稲田大学, 理工学術院, 主任研究員 (90596975)
• Research Collaborator: SOLONNIKOV Vsevolod St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences, Professor ; GALDI Giovanni University of Pittsburgh, Department of Mechanical Engineering and Material Sciences, Department of Mathematics, Lighton E. and Mary N. Orr Professor of Engineering, Professor of Mathematics ; HIEBER Matthias TU Darmstadt, Department of Mathematics, Professor ; ZAJACZKOWSKI Wojciech Institute of Mathematics, Polish Academy of Sciences, Professor ; SCHONBECK Maria University of California Santa Cruz, Department of Mathematics, Professor ; DENK Robert Konstanz University, Department of Mathematics and Statistics, Professor
• Project Period (FY): 2012-05-31 – 2017-03-31
• Keywords: 関数方程式 / 流体数学 / 確率解析 / 大域解析学 / 数値解析

Outline of Final Research Achievements

In our macroscopic studies on mathematical fluid dynamics, we proved the unique existence theorem of locally in time solution of free boundary problems for the Navier-Stokes equations in general domains, employing the theory based on the R boundedness. The unique existence of globally in time solutions and their asymptotic behavior of free boundary problems for the Navier-Stokes equations in both bounded and unbounded domains are also proved based on the spectral analysis of the Stokes operator. In mesoscopic studies, a stochastic differential equation for oscillations of a bubble is derived and analyzed to obtain the unique global solution and its asymptotic behavior. Numerical simulations are also performed based on analysis mentioned above. We developed the theory of Dirac reduction and applied it to Rivlin-Ericksen fluids aiming to formulate a variational principle of fluid dynamics. The Lagrange-Galerkin method was developed and utilized to simulate a rising bubble.

Free Research Field

基礎解析学