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Publications [#303560] of Marc D. Ryser

Papers Published

  1. Ryser, MD; Nigam, N; Tupper, PF, On the well-posedness of the stochastic Allen-Cahn equation in two dimensions (2011) [arXiv:1104.0720v4], [1104.0720v4], [doi]
    (last updated on 2017/12/16)

    Abstract:
    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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