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

该项目主要研究应用科学中非线性模型的建模、分析和计算的基本挑战，包括双曲平衡定律、动力学模型以及简单和复杂流体的混合。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，流体动力学和扩散极限以及规则性方面。(b)多维系统的研究，重点是守恒定律的双曲系统和具有色散效应的可压缩流体模型。要解决的问题包括物理系统的大数据适定性理论，解决方案结构的研究以及奇异极限的分析。这一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。这一领域的解析方法和数值方法是共同发展起来的；分析理解有助于构建准确高效的数值格式，而数值实验往往引导理论分析。这项研究计划的进展将有助于成功研究生物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，该项目将通过为年轻研究人员提供指导和高级培训以及组织与相关领域的研究人员进行跨学科互动的框架，为科学劳动力的发展做出贡献。

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.