Hyperbolic problems in fluids and relativity

Thesis

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.

Authors: Schrecker, M

• DOI: 10.5287/ora-jn1av8e0r
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Spherical symmetry; Vanishing viscosity limit; Entropy solutions; Compensated Compactness; Finite energy method; Transonic nozzle; Euler equations
• Subjects: Hyperbolic Partial Differential Equations; Differential equations, Partial; Special Relativity; Fluid Dynamics