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

非线性偏微分方程（NPDEs）是许多科学进步的核心，其长度尺度从亚原子到天文，时间尺度从皮秒到千年。稳定性分析在npde及其科学和工程应用的各个方面都是至关重要的，但也面临着巨大的挑战。例如，当一个平面激波迎头撞击一个楔子时，当原始激波及时向前移动时，一个自相似的反射激波向外移动。1878年恩斯特·马赫（Ernst Mach）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，可能会出现各种形式的反射-衍射激波。激波反射-衍射的大多数基本问题还没有被理解。在空气动力学中广泛应用的可压缩欧拉系统和势流方程的框架下，激波反射-衍射解的整体存在性和稳定性将是一个明确的数学答案。另一个例子出现在平均场极限的分析中，这是应用分析中的一个强大工具，用于连接许多身体系统的微观和宏观描述。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加流行。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、经济、金融甚至社会科学。一个关键问题是如何通过量化平均场极限和/或它们的水动力近似的稳定性来降低这种复杂性。通过形成一个独特的英美专家联合力量，拟议的研究将解决NPDEs最困难和长期存在的稳定性问题，包括在守恒定律、动力学方程的双曲系统中的渐近、量化和结构稳定性问题，以及跨音速/粘-无粘/流体-颗粒模型中的相关多尺度应用。通过这种横跨大西洋的技能和方法的罕见结合，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涉及稳定/不稳定的新出现的基本问题有关：激波反射/衍射型的稳定性分析，重点研究了气体动力学中最基本的多维激波问题之一——激波反射-衍射问题目标2。涡旋片的稳定性分析，接触不连续，和其他特征不连续的M-D双曲系统的守恒定律，特别是包括欧拉坐标系下的M-D非等熵热弹性方程，控制热弹性非导体的演化；目标3。多智能体系统中粒子对连续统极限的稳定性分析，包括两两相互作用的量化渐近/平均场/大时间极限和一般相互作用的粒子极限；目标4。大初始数据下M-D可压缩黏性流到无黏性流解渐近极限的稳定性分析，着重于黏性极限的消失。这些目标要求很高，因为所涉及的问题是混合类型和多尺度，以及M-D，非局部和非正则性，使得数学分析成为一项艰巨的任务。虽然项目中的许多问题已经为人所知有一段时间了，但直到最近，它们的解决方案似乎才触手可及；事实上，在2010年之前，这个项目的一部分是不可想象的。同时研究与上述四个目标有关的问题将导致对多尺度应用的npde进行更系统的稳定性分析。

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.