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

这项研究项目集中在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。正在研究的模型包括用于研究连续介质物理中出现的非线性系统的双曲平衡定律，用于研究流体-颗粒和流固相互作用的流体混合物，以及用于研究代理人集体自组织的动力学模型。具体主题包括：(A)双曲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极限和流体动力学极限。(C)聚合物流体的大数据存在理论。(D)自组织动力学：捕捉集群行为的模型的长期行为和稳定性。直接解决流体-颗粒相互作用和代理人集体自组织的数学模型与各种问题非常相关：生物的和非生物的(社会网络)。这项调查的进展将增进我们对复杂系统的理解。此外，拟议活动的圆满完成将有助于开发新的分析和数值技术，并有助于设计高性能计算算法。该项目包括一个重要的教育部分，反映了对各级教育(本科生、研究生和研究生教育)的持续承诺。特别关注推广活动、少数民族，以及制定促进和鼓励应用数学领域的培训的方案。

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.