DMS-EPSRC: Stability Analysis for Nonlinear Partial Differential Equations across Multiscale Applications

DMS-EPSRC：跨多尺度应用的非线性偏微分方程的稳定性分析

基本信息

批准号：
EP/V051121/1
负责人：
Gui-Qiang George Chen
金额：
$ 76.73万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV051121%2F1
关键词：
DMS EPSRC Stability Analysis Nonlinear

项目摘要

Nonlinear partial differential equations (NPDEs) are at the heart of many scientific advances, with both length scales ranging from sub-atomic to astronomical and timescales ranging from picoseconds to millennia. Stability analysis is crucial in all aspects of NPDEs and their applications in Science and Engineering, but has grand challenges. For instance, when a planar shock hits a wedge head on, a self-similar reflected shock moves outward as the original shock moves forward in time. The complexity of shock reflection-diffraction configurations was reported by Ernst Mach in 1878, and later experimental, computational, and asymptotic analysis has shown that various patterns of reflected-diffracted shocks may occur. Most fundamental issues for shock reflection-diffraction have not been understood. The global existence and stability of shock reflection-diffraction solutions in the framework of the compressible Euler system and the potential flow equation, widely used in Aerodynamics, will be a definite mathematical answer.Another example arises in the analysis of mean field limits, a powerful tool in applied analysis introduced to bridge microscopic and macroscopic descriptions of many body systems. They typically involve a huge number of individuals (particles), such as gas molecules in the upper atmosphere, from which we want to extract macroscopic information. Multi-agent systems have become more popular than ever. In addition to their new classical applications in Physics, they are widely used in Biology, Economy, Finance, and even Social Sciences. One key question is how this complexity is reduced by quantifying the stability of the mean field limit and/or their hydrodynamic approximations.By forming a distinctive joint force of the UK/US expertise, the proposed research is to tackle the most difficult and longstanding stability problems for NPDEs across the scales, including asymptotic, quantifying, and structural stability problems in hyperbolic systems of conservation laws, kinetic equations, and related multiscale applications in transonic/viscous-inviscid/fluid-particle models. Through this rare combination of skills and methodology across the Atlantic, the project focuses on four interrelated objectives, each connected either with challenging open problems or with newly emerging fundamental problems involving stability/instability:Objective 1. Stability analysis of shock wave patterns of reflections/diffraction with focus on the shock reflection-diffraction problem in gas dynamics, one of the most fundamental multi-dimensional (M-D) shock wave problems;Objective 2. Stability analysis of vortex sheets, contact discontinuities, and other characteristic discontinuities for M-D hyperbolic systems of conservation laws, especially including the equations of M-D nonisentropic thermoelasticity in the Eulerian coordinates, governing the evolution of thermoelastic nonconductors of heat; Objective 3. Stability analysis of particle to continuum limits including the quantifying asymptotic/mean-field/large-time limits for pairwise interactions and particle limits for general interactions among multi-agent systems;Objective 4. Stability analysis of asymptotic limits with emphasis on the vanishing viscosity limit of solutions from M-D compressible viscous to inviscid flows with large initial data.These objectives are demanding, since the problems involved are of mixed-type and multiscale, as well as M-D, nonlocal, and less regular, making the mathematical analysis a formidable task. While many of the problems in the project have been known for some time, it is only recently that their solutions seem to have come within reach; in fact, part of the project would have been inconceivable prior to 2010. The simultaneous study of problems associated with the four objectives above will lead to a more systematic stability analysis for NPDEs across multiscale applications.

非线性偏微分方程（NPDE）是许多科学进步的核心，两个长度尺度从亚原子到天文学，并且从picseconds到几千年的时间表。稳定性分析在NPDE的各个方面及其在科学和工程中的应用至关重要，但面临着巨大的挑战。例如，当平面冲击撞到楔形时，随着原始冲击的时间向前移动，自相似的反射冲击会向外移动。恩斯特·马赫（Ernst Mach）在1878年报道了冲击反射划分配置的复杂性，后来的实验，计算和渐近分析表明，可能发生了各种反射截面冲击的模式。尚未理解大多数冲击反射分配的基本问题。电击反射划分解决方案的全球存在和稳定性在可压量的Euler系统的框架中，并且在空气动力学中广泛使用的潜在流动方程将是一个明确的数学答案。其他示例是对平均场限制的分析，这是一种强大的工具，在应用分析中引入了桥接显微镜和巨型系统系统的应用分析。它们通常涉及大量个体（颗粒），例如高层大气中的气体分子，我们希望从中提取宏观信息。多代理系统比以往任何时候都变得更加受欢迎。除了在物理学方面的新古典应用外，它们还广泛用于生物学，经济，金融甚至社会科学。一个关键的问题是如何通过量化平均场限制和/或其流体动力近似的稳定性来降低这种复杂性。通过形成英国/美国专业知识的独特联合力量，该拟议的研究是解决跨量表的NPD的最困难和长期稳定性问题，包括跨量表，包括在范围内的范围，定量和结构性稳定性的范围，量身定期，量身定量，量化，量化，量化和结构性稳定性。跨性别/粘性渗透/流体粒子模型。通过这种罕见的大西洋技能和方法论的组合，该项目集中于四个相互关联的目标，每个目标都与挑战性的开放问题或新出现的基本问题有关，涉及稳定性/不稳定的基本问题：目标1。对反射的冲击波模式的稳定性分析，重点是震动反射反射的问题，是对震动反射反射问题的重点，是跨性别的问题，是一种跨性别的问题，一项跨度的跨度词，一项跨度的跨度，一项跨度词，一项跨度的跨度，一项跨度的跨度，一项跨度的跨度率（Multive of Indernaltial of Forgutients）。分析涡流纸，接触不连续性和其他特征性不连续性，用于M-D的保护定律系统，尤其是在Eulerian坐标中的M-D非分离热弹性的方程，管理热弹性非导导体的热弹性的演变；目标3。粒子对连续限制的稳定性分析，包括对成对相互作用的渐近/均值/大型/大时间限制，用于成对相互作用和多个代理系统之间的一般相互作用的粒子限制；目标4。稳定性分析，重点是渐近限制的渐近限制，这些范围涉及大量粘度的粘度限制，涉及大量的初始数据。混合型和多尺度，以及M-D，非本地且较少的规则，使数学分析成为一项强大的任务。尽管该项目中的许多问题已经知道了一段时间，但直到最近，他们的解决方案才能实现。实际上，该项目的一部分在2010年之前将是不可想象的。与上述四个目标相关的问题的同时研究将导致对多尺度应用程序的NPDE进行更系统的稳定性分析。