ON THE DYNAMICS OF CERTAIN NONLINEAR SYSTEMS IN APPLIED SCIENCES: TRANSPORT, MOTION AND MIXING

该项目侧重于应用科学中非线性模型的建模，分析和计算的基本挑战，涉及双曲平衡定律，动力学模型以及简单和复杂流体的混合物。该项目有两个主要组成部分：（a）研究复杂流动，重点是流体-颗粒相互作用模型和聚合物流体。要解决的问题包括物理系统的大数据存在理论，稳定性和渐近分析问题，流体动力学和扩散限制以及规律性方面。(b)多维系统的研究，重点是双曲守恒律系统和具有色散效应的可压缩流体模型。这些问题包括物理系统的大数据适定性理论、解的结构研究以及奇异极限分析等，这些跨学科研究的进展将在很大程度上依赖于非线性偏微分方程新分析技术的发展。在这一领域的分析和数值方法一起发展;分析的理解有助于准确和有效的数值方案的建设，而数值实验往往导致理论分析。该研究计划的进展将有助于成功调查生物学和材料科学中出现的各种重要物理系统，以及高性能计算算法的设计。 此外，该项目还将通过为年轻研究人员提供指导和高级培训，并通过组织一个与相关领域研究人员进行跨学科互动的框架，促进科学工作者队伍的发展。

This project focuses on the fundamental challenges of modeling, analysis and computation for nonlinear models in applied sciences involving hyperbolic balance laws, kinetic models, and mixtures of simple and complex fluids. The project has two primary components: (a) The investigation of complex flows with emphasis on fluid-particle interaction models and polymeric fluids. Issues to be addressed include large-data existence theory for physical systems, issues of stability and asymptotic analysis, hydrodynamic and diffusive limits as well as aspects of regularity. (b) The investigation of multidimensional systems with focus on hyperbolic systems of conservation laws and models of compressible fluids with dispersive effects. Issues to be addressed include large-data well-posedness theory for physical systems, the investigation of the structure of solutions as well as the analysis of singular limits.Advances in this interdisciplinary research will rely substantially on the development of new analytical techniques in nonlinear partial differential equations. Analytical and numerical methods in this field have developed together; analytical understanding contributes to the construction of accurate and efficient numerical schemes, while numerical experiments often lead the theoretical analysis. Progress in this research program will contribute to the successful investigation of a wide variety of important physical systems arising in biology and materials science as well as to the design of high performance computational algorithms. Furthermore, this project will contribute to the development of the scientific workforce by providing mentoring and advanced training for young researchers and by organizing a framework for interdisciplinary interactions with researchers from related fields.