双曲空间上的Landau-Lifshitz-Gilbert方程解的全局存在性与自相似爆破解

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 461-474.

双曲空间上的Landau-Lifshitz-Gilbert方程解的全局存在性与自相似爆破解

钟澎洪^1,*(),杨干山²,马璇³

¹ 广东第二师范学院数学系广州 510640
² 云南民族大学数学系昆明 650031
³ 云南师范大学数学科学学院昆明 650092

收稿日期:2017-10-19 出版日期:2019-06-26 发布日期:2019-06-27
通讯作者: 钟澎洪 E-mail:gzydshang@126.com
基金资助:
国家自然科学基金青年基金(11601092);广东省青年创新人才项目基金(2014KQNCX228);广东省科技厅博士启动基金(2014A030310330);广州市科技计划项目基金(201607010352)

Global Existence and Self-Similar Blowup of Landau-Lifshitz-Gilbert Equation on Hyperbolic Space

Penghong Zhong^1,*(),Ganshan Yang²,Xuan Ma³

¹ Department of Mathematics, Guangdong University of Education, Guangzhou 510640
² Department of mathematics, Yunnan Nationalities University, Kunming 650031
³ Institute of Mathematics Science, Yunnan Normal University, Kunming 650092

Received:2017-10-19 Online:2019-06-26 Published:2019-06-27
Contact: Penghong Zhong E-mail:gzydshang@126.com
Supported by:
the National Science Foundation for Young Scientists of China(11601092);the Project for Young Creative Talents of Ordinary University of Guangdong Province(2014KQNCX228);the PhD Start-up Fund of Natural Science Foundation of Guangdong Province(2014A030310330);the Funds of Guangzhou Science and Technology(201607010352)

摘要/Abstract

摘要：

应用Hasimoto变换，给出了双曲空间$\mathbb{H}$²上的Landau-Lifshitz-Gilbert（LLG）方程的一等价系统.基于该等价模型，证明了在小初值条件下LLG方程解的全局存在性.到目前为止，还未见到有文章在双曲空间下给出带阻尼项方程的精确解.基于导出的等价方程，首次构造了一显式小初值的整体解.另外，也给出了等价系统的自相似有限时间爆破解.在作者发表的论文[25]中，构造了在$\mathbb{H}$²上没有吉尔伯特阻尼项方程的有限时间爆破解.带阻尼项的LLG方程的有限能量解能否在$\mathbb{H}$²上演化出有限时间爆破或全局光滑这一问题尚不清楚.该文给出的自相似有限时间爆破解是在整个空间区域上的有限能量解.该例子给出了这个问题的一个回答.

关键词: Landau-Lifshitz-Gilbert方程, 全局存在性, 小解, 爆破解

Abstract:

By the generalized Hasimoto transformation, we deduce an equivalent system of the Landau-Lifshitz-Gilbert equation on hyperbolic space $\mathbb{H}$². Based on this equivalent model, we prove the global existence of the Landau-Lifshitz-Gilbert equation with the small initial condition. Until now, we have not seen a paper discussing the explicit dynamic solution of the complete equation with a damping term on this target. We construct an explicit small data global solution by the equivalent system obtained in this paper. An self-similar finite blowup solution is also presented for the equivalent system. In the previous paper[25], we constructed a finite time blowup solution without Gilbert damping on $\mathbb{H}$². The question of whether a solution of the complete equation with a Gilbert term can develop a finite time blowup from $\mathbb{H}$² and smooth initial data is not clear. The self-similar finite time blowup solution we presented here is a finite energy solution on the entire spacial domain. Our result gives a positive answer to this question.

Key words: Landau-Lifshitz-Gilbert equation, Global existence, Small solution, Blowup solution

中图分类号:

O175.29

钟澎洪,杨干山,马璇. 双曲空间上的Landau-Lifshitz-Gilbert方程解的全局存在性与自相似爆破解[J]. 数学物理学报, 2019, 39(3): 461-474.

Penghong Zhong,Ganshan Yang,Xuan Ma. Global Existence and Self-Similar Blowup of Landau-Lifshitz-Gilbert Equation on Hyperbolic Space[J]. Acta mathematica scientia,Series A, 2019, 39(3): 461-474.

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