On the well-posedness of the stochastic Allen-Cahn equation in two dimensions

Authors:
Marc D. Ryser;Nilima Nigam;Paul F. Tupper
Affiliations:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6 and Department of Mathematics, Simon Fraser University, 8888 University Dr ...;Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada V5A 1S6;Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada V5A 1S6
Venue:
Journal of Computational Physics
Year:
2012

Citing 3
Cited 0

Stochastic Cahn-Hilliard equation

Nonlinear Analysis: Theory, Methods & Applications
An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations

SIAM Review
The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	31.45

Visualization

Abstract

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical systems in space dimensions d=1, 2, 3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d=2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen-Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that shrinking the mesh size in simulations of the two-dimensional white noise-driven Allen-Cahn equation does not lead to the recovery of a physically meaningful limit.