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A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation. / Andrew Neate

Swansea University Author: Andrew Neate

Abstract

This thesis presents a one dimensional analysis of the singularities of the d- dimensional stochastic Burgers equation using the 'reduced action function'. In particular, we investigate the geometry of the caustic, the Maxwell set and the Hamilton-Jacobi level surfaces, and describe some t...

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Published: 2005
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
URI: https://cronfa.swan.ac.uk/Record/cronfa42800
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Abstract: This thesis presents a one dimensional analysis of the singularities of the d- dimensional stochastic Burgers equation using the 'reduced action function'. In particular, we investigate the geometry of the caustic, the Maxwell set and the Hamilton-Jacobi level surfaces, and describe some turbulent phenomena. Chapter 1 begins by introducing the stochastic Burgers equation and its related Stratonovich heat equation. Some earlier geometric results of Davies, Truman and Zhao are presented together with the derivation of the reduced action function. In Chapter 2 we present a complete analysis of the caustic in terms of the derivatives of the reduced action function, which leads to a new method for identifying the singular (cool) parts of the caustic. Chapter 3 investigates the spontaneous formation of swallowtails on the caustic and Hamilton-Jacobi level surfaces. Using a circle of ideas due to Arnol'd, Cayley and Klein, we find necessary conditions for these swallowtail perestroikas and relate these conditions to the reduced action function. In Chapter 4 we find an explicit formula for the Maxwell set by considering the double points of the level surfaces in the two dimensional polynomial case. We extend this to higher dimensions using a double discriminant of the reduced action function and then consider the geometric properties of the Maxwell set in terms of the pre-Maxwell set. We conclude in Chapter 5 by using our earlier work to model turbulence in the Burgers fluid. We show that the number of cusps on the level surfaces can change infinitely rapidly causing 'real turbulence' and also that the number of swallowtails on the caustic can change infinitely rapidly causing 'complex turbulence'. These processes are both inherently stochastic in nature. We determine their intermittence in terms of the recurrent behaviour of two processes derived from the reduced action.
Keywords: Mathematics.
College: Faculty of Science and Engineering

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