The Allen-Cahn equation with generic initial datum
type: Item
instance of: scholarly article
title: The Allen-Cahn equation with generic initial datum
zbMATH Open document ID: 1521.60027
arXiv ID: 2201.08426
publication date: 2023-06-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/2201.08426
review text: The authors consider the Allen-Cahn equation \(\partial_{t}u- \Delta u = u- u^{3}\) with a rapidly mixing Gaussian field as an initial condition. They study the scaling limit of such an equation and show that after a suitably long time the dynamics are well approximated by a certain class of Gaussian nodal sets which evolve under a mean curvature flow. While ideally one would like to consider the white noise initial data, this is not possible in this setting since the scaling exponent of the white noise is below a critical exponent for which one might expect any form of a local well-posedness for the equation. The authors show that if the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to a mean curvature flow.
reviewed by: Maria Gordina
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H40
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 53E10
zbMATH DE Number: 7698588
zbMATH Keywords: Allen-Cahn equation
zbMATH Keywords: white noise
zbMATH Keywords: mean curvature flow
zbMATH Keywords: coarsening
MaRDI profile type: Publication
OpenAlex ID: W4362717973
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DOI: 10.1007/S00440-023-01198-5
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 7698588