Geometric analysis of dispersive flows

2015 Fiscal Year Final Research Report

• Project/Area Number: 24740090
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kochi University
• Principal Investigator: ONODERA Eiji 高知大学, 教育研究部自然科学系理学部門, 准教授 (70532357)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 分散型偏微分方程式 / 初期値問題の時間局所解の存在と一意性 / 渦糸 / 古典スピン系

Outline of Final Research Achievements

The initial value problem for a fourth order dispersive partial differential equation for closed curve flow on a Kaehler manifold was mainly investigated.
The equation models the motion of a vortex filament or the continuum limit of a classical Heisenberg spin chain systems, where the manifold is the two-dimensional unit sphere. In our research, we investigated the relationship between the setting for the manifold and the structure of the equation, and applied the observation to the proof of the existence and the uniqueness of the solution. The main results is the time-local existence and the uniqueness of a smooth solution under the case where the manifold is a closed Riemann surface with constant curvature. Indeed, we found a nice solvable structure of the equation, and completed the proof by the mix of a suitable gauge transformation and the geometric energy method to eliminate the loss of derivatives.

Free Research Field

偏微分方程式、幾何解析