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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
first_indexed 2018-08-02T18:55:34Z
last_indexed 2019-10-21T16:48:29Z
id cronfa42800
recordtype RisThesis
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spelling 2018-08-14T12:05:38.7659398 v2 42800 2018-08-02 A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation. b124512f33eeff87d6ec8760e972335d NULL Andrew Neate Andrew Neate true true 2018-08-02 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. E-Thesis Mathematics. 31 12 2005 2005-12-31 COLLEGE NANME Mathematics COLLEGE CODE Swansea University Doctoral Ph.D 2018-08-14T12:05:38.7659398 2018-08-02T16:24:30.5077956 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Andrew Neate NULL 1 0042800-02082018162522.pdf 10807576.pdf 2018-08-02T16:25:22.8770000 Output 6760692 application/pdf E-Thesis true 2018-08-02T16:25:22.8770000 false
title A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation.
spellingShingle A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation.
Andrew Neate
title_short A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation.
title_full A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation.
title_fullStr A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation.
title_full_unstemmed A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation.
title_sort A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation.
author_id_str_mv b124512f33eeff87d6ec8760e972335d
author_id_fullname_str_mv b124512f33eeff87d6ec8760e972335d_***_Andrew Neate
author Andrew Neate
author2 Andrew Neate
format E-Thesis
publishDate 2005
institution Swansea University
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description 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.
published_date 2005-12-31T19:29:29Z
_version_ 1821344393587589120
score 11.04748

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