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Title: Hyperbolic problems in fluids and relativity
Author: Schrecker, Matthew
ISNI:       0000 0004 7430 6299
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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In this thesis, we present a collection of newly obtained results concerning the existence of vanishing viscosity solutions to the one-dimensional compressible Euler equations of gas dynamics, with and without geometric structure. We demonstrate the existence of such vanishing viscosity solutions, which we show to be entropy solutions, to the transonic nozzle problem and spherically symmetric Euler equations in Chapter 4, in both cases under the simple and natural assumption of relative finite-energy. In Chapter 5, we show that the viscous solutions of the one-dimensional compressible Navier-Stokes equations converge, as the viscosity tends to zero, to an entropy solution of the Euler equations, again under the assumption of relative finite-energy. In so doing, we develop a compactness framework for the solutions and approximate solutions to the Euler equations under the assumption of a physical pressure law. Finally, in Chapter 6, we consider the Euler equations in special relativity, and show the existence of bounded entropy solutions to these equations. In the process, we also construct fundamental solutions to the entropy equations and develop a compactness framework for the solutions and approximate solutions to the relativistic Euler equations.
Supervisor: Chen, Gui-Qiang Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Hyperbolic Partial Differential Equations ; Fluid Dynamics ; Special Relativity ; Differential equations, Partial ; Spherical symmetry ; Transonic nozzle ; Compensated Compactness ; Entropy solutions ; Finite energy method ; Euler equations ; Vanishing viscosity limit