Study on collision phenomena of blow-up points appear in Liouville systems and those of vortices

刘维尔系统中爆炸点与涡旋碰撞现象的研究

• 批准号: 19540222
• 负责人: OHTSUKA Hiroshi
• 金额: $ 2万
• 依托单位: University of Miyazaki
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2009
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19540222/
• 关键词: 関数方程式論; 関数解析学; 応用数学; 数理物理; Liouville system; 変分法; 渦点; 点渦; 関数方程式論; 関数解析学; 応用数学; 数理物理; Liouville system; 変分法; 渦点; 点渦

I studied the mean field equation of vortices with mixed intensities, which is one important example of Liouville systems and got "a mass identity" as a basic fact to classify the collisions of the blow-up points. In addition, I established technique to estimate the Morse index of the critical point about the certain functional that could not assume the Palais-Smale condition including the functionals that lead Liouville systems. Furthermore, I showed a property about the structure around the critical point called "the asymptotic non-degeneracy" about the functionals relating to the scalar Liouville systems (i.e., the Liouville type equations).

我研究了具有混合强度的涡流的平均场方程，这是liouville系统的一个重要例子，并获得了“质量身份”，作为对爆炸点的碰撞进行分类的基本事实。此外，我还建立了技术，以估算某些功能的关键点的摩尔斯索引，该功能无法假设palais-smale条件，包括领导Liouville Systems的功能。此外，我展示了有关与标量liouville系统有关的功能（即liouville类型方程）的临界点周围结构的属性。