Geodesic
On the entropy of parabolic Allen–Cahn equation
Ao Sun1
1Massachusetts Institute of Technology, Cambridge, USA; current affiliation: University of Chicago, USA
Interfaces and free boundaries, Tome 23 (2021) no. 3, pp. 421-432

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DOI

We define a local (mean curvature flow) entropy for Radon measures in Rn or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen–Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen–Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit measure of the parabolic Allen–Cahn equation has unit density for all future time.
DOI : 10.4171/ifb/460
Classification : 53-XX, 35-XX
Mots-clés : Entropy, Allen–Cahn equation, monotonicity formula

Ao Sun  1

1 Massachusetts Institute of Technology, Cambridge, USA; current affiliation: University of Chicago, USA 
Ao Sun. On the entropy of parabolic Allen–Cahn equation. Interfaces and free boundaries, Tome 23 (2021) no. 3, pp. 421-432. doi: 10.4171/ifb/460
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     title = {On the entropy of parabolic {Allen{\textendash}Cahn} equation},
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     year = {2021},
     volume = {23},
     number = {3},
     doi = {10.4171/ifb/460},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/460/}
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