Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application
Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application
复制标题
DOI：
10.1016/j.difgeo.2019.101560
复制
发表时间：
2016-06
影响因子：
0.5
通讯作者：
E. Onodera
中科院分区：
文献类型：
--
作者：
E. Onodera
数学物理领域中出现了一个四阶非线性色散偏微分方程，其解为二维单位球上的曲线流。近十年来，人们考虑了球值物理模型的几何推广，并对初值问题的可解性进行了研究。特别是在作者之前的工作中，在假设解是紧致Riemann曲面上的闭曲线流的情况下，建立了解的时间局部存在唯一性结果。本文提出了球值物理模型的一种新的几何推广方法。在假定初值问题的解是紧致局部厄米对称空间上的闭曲线流的情况下，我们证明了初值问题解的时间局部存在性。该证明是基于几何能量法结合规范变换来克服所谓的导数损失的困难。有趣的是，这些结果可以用于构建Ding和Wang提出的广义bi-Schrödinger流。流形的假设对于初值问题具有良好的可解结构和将广义bi-Schrödinger流动方程简化为本文所考虑的流动方程起着至关重要的作用。
A fourth-order nonlinear dispersive partial differential equation arises in the field of 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 sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrödinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrödinger flow equation to the one considered in the present paper.