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困扰数学家和物理学家的难题。流体一词既包括液体也包括气体，这些流体的流动是由著名的欧拉方程描述的，该方程最初是由莱昂哈德·欧拉在18世纪50年代写下的。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突然发生非常迅速的变化时，冲击波就会发生，数学上将其描述为溶液中的不连续。从那以后，人们发现激波在气体理论中是一种普遍存在的现象，在各种情况下都会自然发生，从排气管或小号中的气体流动到音爆、超新星和爆炸。本奖学金提案旨在研究更广泛的冲击理论中的两个关键问题：冲击反射问题和冲击波的稳定性。当冲击波遇到固体物体并被其反射时，冲击波的反射部分与尚未遇到物体的部分相互作用，就会发生冲击反射。因此，非常简单的初始情景导致了可能非常复杂的气体流动模式。事实上，冲击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，它围绕着一个压力非常高的区域，例如由超新星或爆炸引起的，并膨胀成周围压力低得多的气体。在20世纪40年代，冯·诺伊曼、泰勒和塞多夫的独立研究表明，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微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、物理和几何之间的深层联系开始发挥作用，因为方程的潜在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，即方程所处区域的边界，取决于方程本身的解。奖学金的结果将不仅对数学分析，而且对偏微分方程、数学生物学和几何等领域感兴趣，这些领域的自由边界问题自然而频繁地出现。

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.