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与应用偏微分方程之间的接口问题，重点是分析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决方案。 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流体动力学模型。 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，以及流体力学中的奇异极限。 (b)材料结构的分析，重点是流体-固体相互作用问题以及生物学和环境科学中出现的流体-颗粒模型。 (c)以双曲线平衡定律为重点的无粘性模型研究。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，出现了巨大的空间和时间尺度范围，这使得相关模型的分析具有挑战性。 使用显式积分或甚至数值近似的通用解决方案的控制方程是不切实际的大多数应用。 这项研究探讨了替代方法，这将有助于进一步调查各种重要的物理系统和高性能计算算法的设计。 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，并通过组织一个与相关领域研究人员进行跨学科互动的框架，为科学工作者队伍的发展做出贡献。

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.