Shock Waves in Gas Dynamics

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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.