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The Allen-Cahn equation with generic initial datum

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Martin Hairer
2023

Résumé

We consider the Allen-Cahn equation ?(t)u - ?u = u - u(3) with a rapidly mixing Gaussian field as initial condition. We show that provided that 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 mean curvature flow.

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https://infoscience.epfl.ch/entities/publication/cec53577-43e7-4622-b320-efb3ea19de87

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In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward).

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