Compensated compactness methods in the study of compressible fluid flow

Thesis

This thesis comprises an introduction and three subsequent chapters; each focusing on a particular problem, and containing novel results that complement the existing theory. Throughout, we are concerned with the existence of solutions of the governing equations of inviscid compressible fluid mechanics; the Euler equations. As these equations form an archetypal system of conservation laws, we study them using approaches from the theory of hyperbolic differential equations. Specifically, we implement methods of compensated compactness developed by Tartar and Murat.

To begin with, in Chapter 2, we prove the existence of a finite relative energy entropy solution of the one-dimensional compressible Euler equations under the assumption of an approximately isothermal pressure law. In particular, we obtain this solution as the vanishing viscosity limit of solutions of the one-dimensional compressible Navier–Stokes equations. This procedure can in fact be carried out for a more general class of pressure laws, which we call asymptotically isothermal. This is the subject of Chapter 3, where we show the existence of a finite relative energy entropy solution of the Euler equations in this new setting.

The focus of Chapter 4 is a related two-dimensional stationary problem. Therein, we consider the existence of bounded entropy solutions of the steady compressible potential flow equations in two dimensions; the Morawetz problem for transonic flow. We provide partial results for a γ-law gas of index γ ∈ [3,∞). This involves a detailed analysis of the singular ordinary differential equations that arise when considering the Lax entropy pairs of the system in the presence of a vacuum, which had yet to be performed.

Authors: Schulz, S

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford