Creation of new turbulence analysis method by using diffeomorphism groups of Riemannian geometry

2021 Fiscal Year Final Research Report

• Project/Area Number: 18KK0379
• Research Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Hitotsubashi University (2021); The University of Tokyo (2018-2020)
• Principal Investigator: YONEDA Tsuyoshi 一橋大学, 大学院経済学研究科, 教授 (30619086)
• Project Period (FY): 2019 – 2021
• Keywords: 微分同相写像群 / Euler方程式

Outline of Final Research Achievements

We could proceed the study of the incompressible Euler flow by using diffeomorphism groups, especially we could clarify some relationship between conjugate point and Arnold's stability. The meaning of "conjugate point" is that the small scale vortices (as the perturbations) do not break down along the large scale flow. On the other hand, Arnold's stability represents a state in which small scale vortices as perturbations have been clearly disappearing by the large scale shear flow.
This physical interpretation suggests us that the conjugate point and Arnold's stability are contradictory fluid phenomena. From this point of view, we could proceed the corresponding mathematical analysis, by using the diffeomorphism groups approach.

Free Research Field

数理流体力学

Academic Significance and Societal Importance of the Research Achievements

流体の非線形挙動を深く洞察出来得る数学解析を提示できた、という意味合いにおいて、本研究の学術的意義は大きい。特に、2021年にJFMに出版されたMatsumoto-Otsuki-Ooshida-Gotoの論文で「Euler座標とLagrange座標の違いで乱流の或る重要な統計量が本質的に変わってしまう」ことが物理的に示されており、それは「流体の非線形挙動に対する数学的洞察を飛躍させるためには、Lagrange座標に密接に関係するリー代数構造を深くみていく必要がある」と翻訳出来得る。その問いへの答えを目指す形で「微分同相写像群によるオイラー流の研究」を推し進めることが出来た。