Geometrically Based Kinetic Approach to Multi-scale Problems

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This research project develops analytical and numerical tools to study multi-scale wave dynamics in two application areas: semi-classical dynamics in Schrodinger equations, and kinetic descriptions for polymer orientation dynamics. For high frequency wave propagation problems, the project will continue to develop effective geometrical partial-differential-equation models to capture field statistics. We will further develop mathematical tools to evaluate underlying physical observables and to reconstruct the original wave field globally. Within this framework, semi-classical convergence beyond caustics is to be established. For polymer orientation dynamics, we will study the isotropic-nematic phase transition in presence of inertial forces, as well as derivation and validation of kinetic models for polymers on manifolds. The research employs novel approaches, such as geometric closure for kinetic transport and kinetic models on tangent bundles, combined with traditional methods, such as asymptotic and direct numerical methods.Many physical problems have multiple temporal and spatial scales that pose tremendous difficulties for mathematical analysis and numerical simulation. This research project will have profound impact on fundamental understanding of high frequency waves and polymer orientation dynamics. One of the objectives is to accurately recover high frequency wave fields around caustics, which is important in many applications such as in seismic imaging. Another objective is to investigate the phase transitions in kinetic models of polymers on manifolds, which can lead to a better understanding of phase segregation in polymeric fluids. The theory under development will be applied to and driven by identified practical applications, and the results of the project will not only increase technical knowledge but will also produce a broader view of the subject. A third objective of the project is to provide training for graduate and undergraduate students involved in carrying out this research.