On the Well-Posedness of the Stochastic Allen-Cahn Equation in Two Dimensions

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Numerical Analysis
Dynamical Systems

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological 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 \geq 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 a series of published numerical studies are problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit.

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Ryser, M. D., Nigam, N., & Tupper, P. F. (2012). On the well-posedness of the stochastic Allen–Cahn equation in two dimensions. Journal of Computational Physics, 231(6), 2537-2550. doi:10.1016/j.jcp.2011.12.002