Math @ Duke

Publications [#303560] of Marc D. Ryser
Papers Published
 Ryser, MD; Nigam, N; Tupper, PF, On the wellposedness of the stochastic AllenCahn equation in two
dimensions
(2011) [arXiv:1104.0720v4], [1104.0720v4], [doi]
(last updated on 2018/01/21)
Abstract: White noisedriven 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 illposed by the mathematics community. We study this discrepancy
on the specific example of the two dimensional AllenCahn 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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