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The sharp interface limit of the stochastic Allen-Cahn equation
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Weber, Simon (Researcher in mathematics) (2014) The sharp interface limit of the stochastic Allen-Cahn equation. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2753536~S1
Abstract
We study the Allen-Cahn equation
ut = ϵ2uxx + f(u) + ϵγW
with an additive noise term ϵγW for small ϵ > 0, and in particular the limit ϵ → 0. This is a reaction-diffusion equation, where f (∙) is the negative derivative of a symmetric double-well potential.
We study this equation in the interval (0, 1) with symmetric boundary conditions, and relatively general initial conditions, for W we take space-time white noise.
Brassesco et al., Funaki and other authors showed (with different boundary conditions) that if we can project the solution of the equation to an energy-optimal deterministic solution with just one zero, then in the sharp interface limit ϵ → 0 of the solution appropriately rescaled in time is a standard Brownian motion.
In this work, we extend these results to a much more general case: We start with fairly general initial conditions, show that after some time we are able to project the solution onto energy-optimal deterministic solutions with finitely many zeroes, after which we derive a semimartingale representation for the interfaces; this representation holds until two interfaces get close to each other and annihilate. In the sharp interface limit ϵ → 0, the appropriately time-rescaled interface position of the solution converges weakly to annihilating independent standard Brownian motions. We also derive an analogous result for smooth noise with trace-class covariance operator, in this case the phenomenon happens on a different timescale than for space-time white noise.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Reaction-diffusion equations | ||||
Official Date: | June 2014 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Hairer, Martin | ||||
Sponsors: | Engineering and Physical Sciences Research Council (EPSRC) | ||||
Extent: | 110 leaves | ||||
Language: | eng |
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