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

本研究项目研究连续介质物理和应用偏微分方程组之间的界面问题，重点分析在描述流体流动和连续介质力学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)可压缩和不可压缩流体的流体动力学模型的研究。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)材料结构分析，侧重于生物和环境科学中出现的流固相互作用问题以及流体颗粒模型。在描述可压缩流体、非线性材料、多相流、天体物理系统和生物系统时出现了巨大的时空尺度，这使得对相关模型的分析具有挑战性。对于大多数应用来说，对控制方程的一般解使用显式积分甚至数值逼近是不切实际的。这项研究探索了其他方法，这些方法将有助于进一步研究各种重要的物理系统，并有助于设计高性能计算算法。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，并通过组织与相关领域的研究人员进行跨学科互动的框架，促进科学队伍的发展。

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.