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Stochastic Partial Differential Equations

Authors: Danial Sanz Alonso, Matthew Dunlop, Graham Hobbs, David O'Connor
Supervisors: Martin Hairer, Hendrik Weber


The Allen-Cahn equation models the evolution of an interface between two different media. It was first introduced in material science to study the behaviour of an interface separating two different iron and aluminium alloys. The following is a picture of the original paper where you can see the two different media.

Different iron and aluminium alloys


Ever since, it has been successfully used to model a wide range of moving interface problems, such as vesicle membranes, nucleation of solids or mixture of two incompressible fluids. You can find more about the deterministic Allen-Cahn equation here.

Our project deals with the stochastic Allen-Cahn equation (SACE), which also models the behaviour of a moving interface. However, it describes in a more realistic way than its deterministic counterpart the evolution of the interface in problems that are inherently random or highly unpredictable, such as the interface between water and ice. Learn more on the stochastic Allen-Cahn equation here.

An overview of our results is given below.

1D Stochastic Allen-Cahn with Asymmetric Double-well Potential

In this section we generalise the main result in Interface Fluctuations and Couplings in the D=1 Ginzburg–Landau Equation with Noise, by S. Brassesco, A. De Masi and E. Presutti, to the case of asymmetric double-well potentials. We show, as conjectured in our research proposal, that now the centre moves according to a Brownian motion with a drift.

Numerics for the 2D Stochastic Allen Cahn Equation

Here some numerical experiments for the two-dimensional equation are presented. The code builds on a one-dimensional version introduced in A finite element method via noise regularization for the stochastic Allen-Cahn problem by M. A. Katsoulakis, G.T. Kossioris, and O. Lakkis.

Acknowledgements

We acknowledge and thank the help of our supervisors Prof Martin Hairer and Dr Hendrik Weber.
We also acknowledge the funding body EPSRC and the support from MASDOC CDT.

Contact: d.sanz-alonso at warwick.ac.uk, matthew.dunlop at warwick.ac.uk,
g.hobbs at warwick.ac.uk, d.g.m.o-connor at warwick.ac.uk

MASDOCEPSRC


The Allen-Cahn equation models the interface evolution between two different media