Analysis on Faddeev, Skyrme and Some Complex Fluid Models

Faddeev、Skyrme和一些复杂流体模型的分析建议研究摘要林芳华该奖项将支持两个不同项目的研究，这两个项目在数学和科学上都是重要和有趣的。其中一个项目是研究Faddeev和Skyrme模型的静态方程和演化方程中的类粒子解。我们还计划研究这些问题的离散类比(或格子模型)，并探索离散模型和连续模型之间的联系。第二个项目将是粘弹性流体流体动力学的理论研究，以及聚合物流体的微观-宏观模型。这些问题涉及多尺度现象，由双曲型和抛物型偏微分方程组的非线性耦合系统来模拟。Skyrme和Faddeev模型使用连续扩展的、拓扑特征的、相对论不变的、局部集中的、类孤子的场来模拟基本粒子。这些粒子状的解显示了高能物理、生物科学和天文学中出现的一些基本现象的特征。事实上，最近的研究表明，Faddeev模型可以捕捉到场结构的打结结构；这是任何其他经典场模型还没有做过的事情。拟议研究的一个主要目标是对静态和动态模型进行系统的数学研究。另一个目标是通过分析其他可能包含许多尺度的自然现象，如复杂流体和生物流体，来研究大尺度场中这些颗粒状解之间的相似之处。

Analysis on Faddeev, Skyrme and some Complex Fluid ModelsAbstract of Proposed ResearchFang-Hua Lin This award will support research on two different projects that are important and interesting both mathematically and scientifically. One project is to study the particle-like solutions in both the static and evolutionary equations of Faddeev and Skyrme models. We also plan to study the discrete analogs of these problems (or lattice-models) and to explore the connections between the discrete and continuous models. The second project will be a theoretical study of the hydrodynamics of viscoelastic fluids, and micro-macro models, of polymeric fluids. These problems involve multiple scale phenomena and are modeled by nonlinear coupled systems of partial differential equations of both hyperbolic and parabolic types. Skyrme and Faddeev models use continuously extended, topologically characterized, relativistically invariant, locally concentrated, soliton-like fields to model elementary particles. These particle-like solutions display the features of some fundamental phenomena that arise in high energy physics, biological sciences and astronomy. Indeed, it was recently shown that Faddeev models can capture the knotted structures of field configurations; something that has not been yet done for any other classical field models. A primary object of the proposed research is to conduct a systematic mathematical study of both static and dynamic models. .Another objective is to investigate the analogies between these particle-like solutions in a large scale field with the analysis of other natural phenomena which may contain many scales such as complex fluids and biological fluids.