Geometrically Based Kinetic Approach to Multi-scale Problems

该研究项目开发了分析和数值工具，以研究两个应用领域的多尺度波动动力学：薛定谔方程中的半经典动力学和聚合物取向动力学的动力学描述。 对于高频波传播问题，该项目将继续开发有效的几何偏微分方程模型，以获取现场统计数据。 我们将进一步开发数学工具来评估潜在的物理观测值，并在全球范围内重建原始波场。 在这个框架内，超越焦散的半经典收敛将被建立。 对于聚合物取向动力学，我们将研究在惯性力存在下的各向同性相变，以及流形上聚合物动力学模型的推导和验证。 由于许多物理问题具有多个时空尺度，给数学分析和数值模拟带来了巨大的困难，因此，本文采用了新的方法，如动力学输运的几何封闭和切丛动力学模型，并结合了传统的方法，如渐近和直接数值方法。 该研究项目将对高频波和聚合物取向动力学的基本理解产生深远的影响。 目标之一是准确地恢复焦散线周围的高频波场，这在诸如地震成像的许多应用中是重要的。 另一个目标是研究歧管上聚合物动力学模型中的相变，这可以更好地了解聚合物流体中的相分离。 正在开发的理论将被应用到确定的实际应用中并由其驱动，该项目的结果不仅将增加技术知识，而且还将产生对该主题的更广泛的看法。 该项目的第三个目标是为参与开展这项研究的研究生和本科生提供培训。

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.