Analysis on Faddeev, Skyrme and Some Complex Fluid Models

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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.