Scott, Michael R. (2014) A parabolic PDE on an evolving curve and surface with finite time singularity. PhD thesis, University of Warwick.

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Abstract

We consider the heat equation

∂∙tu + u∇M(t) . v - ∆M(t)u = 0
u(x,0) = u0
x ∈ M(0)

on an evolving curve which forms a "kink" in finite time. We describe the behaviour of the solution at the singularity and look to continue the solution past the singularity. We perturb the heat equation and study the effects of a deterministic perturbation and a stochastic perturbation on the solution, before the singularity. We then consider the heat equation on an evolving surface that forms a "cone" singularity in finite time and study the behaviour of the solution at the singularity. We then look to continue the solution past the singularity, in some probabilistic sense. Finally, we consider the heat equation on an evolving curve, where the evolution of the curve is coupled to the solution of the equation on the curve. We prove existence and uniqueness of the solution for small times, before any singularity can occur.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Heat equation, Differential equations, Parabolic, Singularities (Mathematics)
Official Date:	September 2014
Dates:	Date Event September 2014 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics and Statistics Doctoral Training Centre
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Hairer, Martin
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC) [EP/HO23364/1]
Extent:	vi, 176 leaves : charts
Language:	eng

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