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Title: Transonic shock waves and free boundary problems for the nonlinear wave system
Author: Rigby, Matthew
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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In this Thesis, we study the problem of Shock Reflection-Diffraction by a Wedge for the Nonlinear Wave System. Shock Reflection-Diffraction problems in Fluid Dynamics, and the corresponding wave patterns have been studied by many authors, via many different analytical, experimental and numerical methods. The Nonlinear Wave System is obtained from the Full Euler System, by making the assumption that the inertial terms are small. It has many similar properties (such as the structure of the characteristics) in common with it, so is a useful model for understanding the Full Euler System, and also plays an important role in some operator splitting schemes in numerical analysis. The solutions to the problem of Shock Reflection-Diffraction by a wedge that we seek are Self-Similar Regular Shock Reflection-Diffraction configurations. The first necessary ingredient in the structure of these solutions is the existence of local reflection at the point where the incident shock hits the wedge. We find that there exists sonic and detachment angles, depending only on the initial data, determining the existence of the local reflection, and the pattern which appears. With the local theory in hand, we use a similar approach to that used in [1, 26], by using a nonlinear method of continuity that relies on the Leray- Schauder degree theory from [87], to obtain existence in all cases where the local reflection exists. There are three key difficulties in the approach. The first is the changing elliptic or degenerate elliptic structure as the wedge angle passes through the sonic angle, and the corresponding change in wave patterns. In particular, we find that for some subsonic wedge angles, we require a new technique to attain suitable regularity near the reflection point, from the limited structure estimates available. The second is the degenerate obliqueness of the free boundary condition where the shock wave meets the flat part of the wedge. Because obliqueness is a property that relies not just on the values of the coeficients, but on the shape of the boundary, this is handled via a new technique involving imposing a cutoff on the boundary itself. The third is ensuring that, in the iteration problem, we obtain a gainin- regularity so that compactness of the iteration map holds. This involves careful analysis of the regularity of both the density and boundary.
Supervisor: Chen, Gui-Qiang G. Sponsor: Engineering and Physical Sciences Research Council (EPSRC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available