Partial differential equation: Schrodinger operator and long-time dynamics

偏微分方程：薛定谔算子和长期动力学

• 批准号: FT230100588
• 负责人: A/Prof Zihua Guo
• 金额: $ 72.6万
• 依托单位: Monash University
• 依托单位国家: 澳大利亚
• 项目类别: ARC Future Fellowships
• 财政年份: 2024
• 资助国家: 澳大利亚
• 起止时间: 2024-06-30 至 2028-06-29
• 项目状态: 未结题
• 来源: https://dataportal.arc.gov.au/RGS/Web/Grants/FT230100588
• 关键词: Partial; differential; equation; Schrodinger; operator

This project aims to develop new analysis methods associated to the Schrodinger operator, and to solve several challenging problems regarding dispersive partial differential equations (PDE). Long-time dynamics of PDE solutions are a key goal in both pure and applied mathematics, and have been extensively studied by leading mathematicians and mathematical physicists. However, it is unknown how to investigate large solutions when the order of the PDE's nonlinearity is low. This project expects to develop new methods to attack such problems. The results of the project will be of great importance in mathematics and physics, as many fundamental physical models in areas such as optics, fluid mechanics and quantum mechanics fit the paradigm.

本计画旨在发展与薛定谔算子相关的新分析方法，并解决关于色散偏微分方程（PDE）的几个具有挑战性的问题。偏微分方程解的长时间动力学是纯数学和应用数学中的一个关键目标，并且已经被领先的数学家和数学物理学家广泛研究。然而，当偏微分方程的非线性阶数较低时，如何研究大的解是未知的。该项目希望开发新的方法来解决这些问题。该项目的成果将在数学和物理学中具有重要意义，因为光学、流体力学和量子力学等领域的许多基本物理模型都符合该范式。