Geometrically Based Kinetic Approach to Multi-scale Problems

该研究项目开发了分析和数值工具来研究两个应用领域的多尺度波动动力学：薛定谔方程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来捕获现场统计数据。我们将进一步开发数学工具来评估潜在的物理观测，并在全球范围内重建原始波场。在这个框架内，将建立超越焦散的半经典收敛。对于聚合物取向动力学，我们将研究惯性力存在下的各向同性向列相转变，以及流形上聚合物动力学模型的推导和验证。该研究采用了新的方法，如动力学输运的几何闭合和切线束上的动力学模型，结合了传统的方法，如渐近和直接数值方法。许多物理问题具有多个时间和空间尺度，这给数学分析和数值模拟带来了巨大的困难。该研究项目将对高频波和聚合物取向动力学的基础认识产生深远的影响。其中一个目标是准确地恢复焦散周围的高频波场，这在地震成像等许多应用中都很重要。另一个目标是研究聚合物在流形上的动力学模型中的相变，这可以更好地理解聚合物流体中的相分离。正在发展的理论将被应用于确定的实际应用，项目的结果不仅会增加技术知识，而且会产生更广泛的主题观点。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.