ON THE DYNAMICS, STRUCTURE AND STABILITY OF CERTAIN NONLINEAR SYSTEMS IN APPLIED SCIENCES

该研究项目的重点是物理和生物科学以及工程应用中出现的非线性系统的建模和数学分析。正在调查的模型包括双曲平衡法的调查中出现的连续物理，混合物的流体的研究流体-颗粒和流体-固体相互作用的非线性系统，以及动力学模型的调查的集体自组织的代理。 （a）双曲线平衡定律和流体混合物模型的解的结构和稳定性。(b)碰撞动力学方程的扩散和流体动力学极限。(c)聚合物流体的大数据存在性理论。(d)自组织动力学：捕捉群集行为的模型的长时间行为和稳定性。直接处理流体-颗粒相互作用和代理的集体自组织的数学模型与各种问题非常相关：生物和非生物（社交网络）。这项研究的进展将增进我们对复杂系统的理解。此外，成功完成拟议的活动将有助于开发新的分析和数值技术，并设计高性能的计算算法。该项目包括一个实质性的教育部分，反映了对各级教育（本科、研究生和研究生教育）的持续承诺。特别注意的是外展活动，少数民族，并制定计划，促进和鼓励在应用数学领域的培训。

This research project focuses on the modeling and mathematical analysis of nonlinear systems arising in physical and biological science as well as engineering applications. The models under investigation include hyperbolic balance laws for the investigation of nonlinear systems arising in continuum physics, mixtures of fluids for the study of fluid-particle and fluid-solid interaction as well as kinetic models for the investigation of the collective self-organization of agents. Specific themes include: (a) Structure and stability of solutions to hyperbolic balance laws and models of fluid mixtures. (b) Diffusion and hydrodynamic limits for collisional kinetic equations. (c) Large-data existence theory for polymeric fluids. (d) Self-organized dynamics: long-time behavior and stability of models capturing flocking behavior. Mathematical models that directly address fluid-particle interaction and collective self-organization of agents are very relevant to a variety of problems: biological, and non-biological (social networks). Progress in this investigation will enhance our understanding of complex systems. Furthermore, a successful completion of the proposed activities will contribute to the development of new analytical and numerical techniques and to the design of high performance computational algorithms. The project includes a substantial educational component that reflects an ongoing commitment to education on all levels (undergraduate, graduate, and post-graduate education). Special attention is given to outreach activities, minorities, and to developing programs that foster and encourage training in the field of applied mathematics.