On the Dynamics of Nonlinear Systems in Applied Sciences: From Theory, Computations, and Experiments to Insights

本研究的重点是在物理应用中出现的非线性偏微分方程组的建模和数学分析。该分析涉及可压缩流动、混合物和聚合物流体的非线性系统，这与科学和工程中的几个实际应用有关，以及描述自组织动力学的动力学模型，这在生物学、物理学和生物工程中很重要。该项目包括为研究生提供培训和研究机会。在流体动力学中，随机驱动系统中统计平稳状态的存在从数学和实验的角度都具有重要意义。一方面，不变测度的存在提供了系统长期行为的一般信息。另一方面，在遍历假设下，它们提供了实验观测之间的联系，例如在湍流理论中。本项目旨在(a)建立确定性系统变分解的适定性结果，(b)分析随机强迫系统的弱鞅解和不变测度。在本研究中开发的变分框架将用于构建用于科学和工程中出现的非线性系统的近似的数值算法。该项目的目标之一是将理论和计算结果与实验联系起来，以优化理论模型，以获得更准确的预测。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research investigation focuses on the modeling and mathematical analysis of nonlinear systems of partial differential equations arising in physical applications. The analysis involves nonlinear systems for compressible flows, mixtures, and polymeric fluids, which are relevant to several practical applications in science and engineering, and kinetic models describing self-organized dynamics, which are important in biology, physics, and bioengineering. The project includes training and research opportunities for graduate students. The existence of statistically stationary states in randomly driven systems in fluid dynamics is of basic importance from both mathematical and experimental points of view. On the one hand, the existence of an invariant measure provides information on the generic long-time behavior of the system. On the other hand, under ergodicity assumptions, they provide a link between experimental observations, for example in turbulence theory. This project aims to (a) establish well-posedness results for variational solutions to deterministic systems and (b) analyze weak martingale solutions and invariant measures for randomly forced systems. The variational framework developed in this investigation will be employed for the construction of numerical algorithms for the approximation of nonlinear systems that arise in science and engineering. One of the goals of the project involves connecting the theoretical and computational results to experiments, in order to optimize theoretical models for more accurate predictions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.