Shock Waves in Gas Dynamics

几个世纪以来，理解流体中冲击波的性质和行为一直是数学家和物理学家的难题。术语流体包括液体和气体，这些流体的流动由著名的欧拉方程描述，该方程首先由Leonhard Euler在1750年代写下。在19世纪40年代，斯托克斯在这些方程的解中发现了一种新现象：激波。当流体的密度或速度在很短的时间内发生突然的、非常迅速的变化时，就会发生冲击波，在数学上被描述为溶液中的不连续性。从那时起，人们发现激波是气体理论中普遍存在的现象，从排气管或喇叭中的气体流动到音爆、超新星和爆炸，都是自然发生的。本奖学金计划旨在研究更广泛的冲击理论中的两个关键问题：冲击波反射问题和冲击波的稳定性。当冲击波遇到固体物体并从其反射时，冲击波的反射部分与尚未遇到物体的冲击波的那些部分相互作用。因此，非常简单的初始场景导致气体流动的潜在非常复杂的模式。事实上，激波反射问题是黎曼问题的一个例子，黎曼问题是欧拉方程解的基本组成部分。另一方面，冲击波是一种球形冲击波，它围绕着一个非常高压的区域，例如由超新星或爆炸引起的，并扩展到周围压力低得多的气体中。在20世纪40年代，冯·诺依曼、泰勒和谢多夫的独立工作确立了这些激波可以用欧拉方程的解来描述，但是这些解对周围气体的小扰动的稳定性的基本问题仍然是一个重要的悬而未决的问题。在这两个问题中，数学、物理学和几何学之间的深刻联系发挥了作用，因为方程的基本对称性和冲击的几何结构相互作用。此外，这两个问题都是自由边界问题，因为激波的位置，即方程成立的区域的边界，取决于方程本身的解。该奖学金的结果将不仅在数学分析中感兴趣，而且在偏微分方程，数学生物学和几何学领域也是如此，其中自由边界问题自然和频繁地发生。

The problem of understanding the nature and behaviour of shock waves in fluids has been a puzzle to mathematicians and physicists for centuries. The term fluids encompasses both liquids and gases and the flow of such fluids is described by the famous Euler equations, first written down by Leonhard Euler in the 1750s. In the 1840s, Stokes discovered a new phenomenon in the solutions of these equations: the shock wave. A shock wave occurs when there is a sudden, very rapid change in the density or velocity of a fluid over a very short space of time, described mathematically as a discontinuity in the solution. Since then, shocks have been discovered to be a ubiquitous phenomenon throughout the theory of gases, occurring naturally in situations ranging from the flow of gas in an exhaust pipe or a trumpet to sonic booms, supernovas and explosions. This Fellowship proposal aims to investigate two key problems within the broader theory of shocks: the shock reflection problem and the stability of blast waves. Shock reflection occurs when a shock wave meets a solid object and is reflected from it, the reflected part of the shock interacting with those parts of the shock that have not yet met the object. Thus the very simple initial scenario leads to a potentially very complex pattern of gas flow. In fact, the shock reflection problem is an example of a Riemann problem, the fundamental building blocks of the solutions to the Euler equations. Blast waves, on the other hand, are spherical shock waves that surround an area of very high pressure, for example caused by a supernova or explosion, and expand into a surrounding gas of much lower pressure. In the 1940s, independent work of von Neumann, Taylor and Sedov established that these shock waves could be described by solutions of the Euler equations, but the fundamental question of the stability of these solutions to small perturbations of the surrounding gas remains a significant open question. In both of these problems, deep connections between mathematics, physics and geometry come into play, as the underlying symmetries of the equations and geometric structures of the shocks interact. Moreover, both of these problems are free boundary problems because the location of the shock, which is a boundary for the region in which the equations hold, depends on the solution of the equations themselves. The results of the Fellowship will be of interest not only in Mathematical Analysis, but in the areas of Partial Differential Equations, Mathematical Biology, and Geometry also, where free boundary problems occur naturally and frequently.