A fourth-order dispersive flow equation for closed curves on compact Riemann surfaces
A fourth-order dispersive flow equation for closed curves on compact Riemann surfaces
复制标题
紧黎曼曲面上闭曲线的四阶色散流方程
DOI：
10.1007/s12220-017-9808-1
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Eiji Onodera
中科院分区：
文献类型：
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作者：
Matsuoka Masayuki;Onodera Eiji;Kawakami Toshitsugu;Takano Kazutaka;Kimura Yuzuru;Shuji Yoshikawa;Shuji Yoshikawa;Eiji Onodera
在物理学和流体力学的某些背景下，出现了正则二维单位球上封闭曲线的四阶色散流动方程。本文考虑了球值模型的一个几何推广，其中的解假定取值于紧致黎曼曲面。作为主要结果，在黎曼曲面的截面曲率为常数的假设下，建立了初值问题解的时间局部存在性和唯一性。分析的困难来自于所谓的导数损失和局部平滑效应的缺乏。证明是基于几何能量方法结合一种规范变换，以消除导数的损失。具体来说，显示的唯一性的解决方案，详细的几何分析的可解结构的方程。
A fourth-order dispersive flow equation for closed curves on the canonical two-dimensional unit sphere arises in some contexts in physics and fluid mechanics. In this paper, a geometric generalization of the sphere-valued model is considered, where the solutions are supposed to take values in compact Riemann surfaces. As a main result, time-local existence and uniqueness of a solution to the initial value problem are established under the assumption that the sectional curvature of the Riemann surface is constant. The analytic difficulty comes from the so-called loss of derivatives and the absence of the local smoothing effect. The proof is based on the geometric energy method combined with a kind of gauge transformation to eliminate the loss of derivatives. Specifically, to show the uniqueness of the solution, the detailed geometric analysis of the solvable structure for the equation is presented.