On the Dynamics, Structure and Stability of Certain Nonlinear Systems in Applied Sciences

本研究项目探讨了连续介质物理与应用偏微分方程之间的界面问题，重点分析了流体流动和连续介质力学描述中出现的非线性系统的物理相关解。该项目有三个主要组成部分：(a)研究可压缩和不可压缩流体的水动力模型。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，以及流体动力学中的奇异极限。(b)对材料结构的分析，重点是流体-固体相互作用问题以及生物和环境科学中出现的流体-颗粒模型。(c)研究无粘模型，重点是双曲平衡定律。在可压缩流体、非线性材料、多相流、天体物理系统和生物系统的描述中出现的空间和时间尺度的巨大范围使得相关模型的分析具有挑战性。对控制方程的一般解使用显式积分或甚至数值近似在大多数应用中是不切实际的。本研究探索了有助于进一步研究各种重要物理系统和设计高性能计算算法的替代方法。该项目的目标之一是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 research project investigates problems on the interface between continuum physics and applied partial differential equations, with emphasis on analysis of physically relevant solutions of the nonlinear systems that arise in the description of fluid flow and continuum mechanics. The project has three primary components: (a) The investigation of hydrodynamic models for compressible and incompressible fluids. Issues to be addressed include large-data existence theory for physical systems, issues of stability and asymptotic analysis, and singular limits in hydrodynamics. (b) The analysis of material structures with focus on fluid-solid interaction problems as well as fluid-particle models arising in biology and environmental sciences. (c) The investigation of inviscid models with emphasis on hyperbolic balance laws.The enormous range of spatial and temporal scales that arise in the description of compressible fluids, nonlinear materials, multi-phase flows, astrophysical systems, and biological systems makes analysis of the associated models challenging. Use of explicit integration or even numerical approximation for generic solutions to the governing equations is impractical for the majority of applications. This research explores alternative approaches that will contribute to the further investigation of a wide variety of important physical systems and to the design of high performance computational algorithms. One of the goals of this project is to 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.