Developments in Geometric Analysis of the initial value problem for dispersive flow equation

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05235
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Kochi University
• Principal Investigator: Onodera Eiji 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): 2016-04-01 – 2020-03-31
• Keywords: 高階分散型偏微分方程式 / ２重シュレーディンガー写像流 / 局所エルミート対称空間 / 時間局所解の一意存在

Outline of Final Research Achievements

A fourth-order nonlinear dispersive partial differential equation arises in mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the physical model has been proposed.In particular, by the present researcher, local existence and uniqueness of a smooth solution to the initial value problem was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In this research, a new geometric generalization of the physical model was introduced. Moreover, local existence of a solution to the initial value problem was obtained under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. In addition, the initial value problem for a fifth-order dispersive equation for curve flow on the sphere was also handled.

Free Research Field

偏微分方程式, 幾何解析

Academic Significance and Societal Importance of the Research Achievements

Ding-Wang(2018, Math.Z)により、リーマン多様体からケーラー多様体への写像流に対する幾何学的偏微分方程式が提案され、その解は(シュレーディンガー写像流の自然な高階版という意味で)一般化2重シュレーディンガー写像流と呼ばれる。本研究の4階の分散型方程式に対する成果は、写像流の定義域が1次元（つまり曲線流）という限定的設定化ではあるが、一般化2重シュレーディンガー写像流の存在を保証した初めての成果と言える。今後は解の一意性や空間高次元化に関する研究への進展が期待される。また、5階の分散型方程式に対する成果は、任意奇数階の分散型方程式の場合への拡張が期待される。