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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:640984 |
| Date | January 2014 |
| Creators | Scott, Michael R. |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/66752/ |
Page generated in 0.0022 seconds