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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，其长度范围从亚原子到天文学，时间范围从皮秒到千年。稳定性分析在非线性偏微分方程的各个方面及其在科学和工程中的应用中都至关重要，但也面临着巨大的挑战。例如，当一个平面激波迎面撞击一个楔形体时，自相似的反射激波会随着原始激波在时间上向前移动而向外移动。马赫在1878年报道了激波反射-衍射构型的复杂性，后来的实验、计算和渐近分析表明，可能出现各种各样的反射-衍射激波。激波反射-衍射的大多数基本问题还没有弄清楚。在可压缩Euler方程和势流方程的框架下，激波反射-绕射解的整体存在性和稳定性将是一个明确的数学答案。另一个例子是平均场极限分析，它是应用分析中的一个强有力的工具，被引入许多物体系统的微观和宏观描述之间的桥梁。它们通常涉及大量的个体（粒子），例如高层大气中的气体分子，我们希望从中提取宏观信息。多智能体系统比以往任何时候都更受欢迎。除了它们在物理学中的新经典应用外，它们还广泛应用于生物学，经济学，金融学甚至社会科学。一个关键问题是如何通过量化平均场极限和/或其流体动力学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，拟议的研究将解决跨尺度NPDE最困难和长期存在的稳定性问题，包括守恒律，动力学方程，以及跨音速/粘性-非粘性/流体-粒子模型中的相关多尺度应用。通过跨大西洋的技能和方法的这种罕见的组合，该项目侧重于四个相互关联的目标，每个目标都与具有挑战性的开放问题或新出现的涉及稳定/不稳定的基本问题有关：目标1。反射/衍射的冲击波模式的稳定性分析，重点是气体动力学中的冲击反射-衍射问题，这是最基本的多维（M-D）冲击波问题之一;目标2。涡面的稳定性分析，接触不连续性，和其他特征的不连续性的M-D双曲守恒律系统，特别是包括方程的M-D非等熵热弹性在欧拉坐标系中，控制热弹性非导体的发展;目标3。粒子到连续极限的稳定性分析，包括成对相互作用的量化渐近/平均场/大时间极限和多智能体系统之间一般相互作用的粒子极限;目标4。渐近极限的稳定性分析，着重于从M-D可压缩粘性流到无粘流的解的粘性极限的消失，这些目标是苛刻的，因为所涉及的问题是混合型和多尺度的，以及M-D，非局部的和不太规则的，使得数学分析成为一项艰巨的任务。虽然该项目中的许多问题已为人们所知有一段时间了，但直到最近，这些问题的解决似乎才触手可及;事实上，该项目的一部分在2010年之前是不可想象的。与上述四个目标相关的问题的同时研究将导致跨多尺度应用的NPDE的稳定性分析。