Lam, King Ming;
(2024)
Stability properties of Lane-Emden and Goldreich-Weber solutions to the Euler-Poisson system.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The Euler-Poisson equation (or system) is the compressible Euler equation (which describes a compressible fluid) coupled with the Poisson equation (which describes Newtonian gravity). In a free boundary context where the domain evolved according to the fluid motion, this system describes a classical model of a star – a lump of compressible gas or liquid surrounded by a vacuum subject to forces created by its own pressure and gravity – and has been long studied by astrophysicists. Two classes of important special solutions to this system are the Lane-Emden stars and the Goldreich-Weber stars. The former is a class of spherically symmetric static solutions describing a static star, while the latter is a class of expanding/collapsing solution which could describe for instance a supernova expansion or gravitational collapse. An important question regarding these solutions are their stability properties under perturbations. The expanding Goldreich-Weber stars consist of two types – one that expands at a linear rate and one that expands at a self-similar rate. In this thesis we prove that the former is non-linearly stable under perturbations (allowed to be non-radial), and the latter class is codimension-1 non-linearly stable under irrotational perturbations (also allowed to be non-radial). In the next part of the thesis, we will establish the linear stability properties of the liquid Lane-Emden stars, in particular we found that it differs from that of gaseous Lane-Emden stars. We establish various qualitative properties of the liquid Lane-Emden stars and using them we show that their linear stability properties depend not only on the adiabatic index but also the central density of the star. Such dependence on central density is not seen in the gaseous Lane-Emden stars.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Stability properties of Lane-Emden and Goldreich-Weber solutions to the Euler-Poisson system |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2024. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| Keywords: | Partial Differential Equations, Analysis of PDEs, Mathematical Physics |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10186072 |
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