The Allen-Cahn equation with generic initial datum
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.
scientific article; zbMATH DE number 7698588