Geometric analysis of higher-order dispersive flows

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03703
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Kochi University
• Principal Investigator: Eiji Onodera 高知大学, 教育研究部自然科学系理工学部門, 准教授 (70532357)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 非線型分散型偏微分方程式

Outline of Final Research Achievements

This research mainly focused on a fourth-order dispersive partial differential equation and the initial value problem for curve flows on a compact K\"ahler manifold. The outline of the research achievements is stated as follows:
(1)We investigated the above initial value problem for closed curve flows on a compact locally Hermitian symmetric space,and proved the uniqueness of a solution for initial data in a Sobolev space with high regularlity.
(2)We investigated the above equation for open curve flows on a compact K\"ahler manifold, and presented a framework that can transform the equation into a system of fourth-order nonlinear dispersive partial differential-integral equations for complex-valued functions, which was achieved by developing the so-called generalized Hasimoto transformation. Moreover, we verified the local-wellposedness of the initial value problem for a related nonlinear system satisfied by complex-valued functions.

Free Research Field

偏微分方程式、幾何解析

Academic Significance and Societal Importance of the Research Achievements

上記成果(1)について：閉曲線流の場合の像空間への設定という意味ではこれ以上の緩和はほぼ不可能と思われる局所エルミート対称性のもとで初期値問題が一意可解であることが確認された。そのために考案した、像空間をユークリッド空間に等長的に埋め込んだときの局所エルミート対称性の利用法は、他の類似的問題への応用も期待される。
上記成果(2)について：像空間が高次元コンパクトケーラー多様体である場合も含めて統一的に扱うことのできる変換法が与えられた。複素次元が２以上のコンパクトケーラー多様体の構造と単独でない非線型４階分散型偏微分方程式系の構造との対応という融合的観点から更なる研究の発展が期待される。