Consider the Allen–Cahn equation on the d-dimensional torus, d = 2, 3, in the sharp interface limit. As is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen–Cahn equation and describe its large deviation asymptotics in a joint sharp interface and small noise limit. Relying on previous results on the variational convergence of the action functional, we prove the large deviations upper bound. The corresponding rate function is finite only when there exists a time evolving interface of codimension one between the two stable phases. The zero level set of this rate function is given by the evolution by mean curvature in the sense of Brakke. Finally, the rate function can be written in terms of the sum of two non-negative quantities: the first measures how much the velocity of the interface deviates from its mean curvature, while the second is due to the possible occurrence of nucleation events.

Stochastic Allen–Cahn approximation of the mean curvature flow: large deviations upper bound / BERTINI MALGARINI, Lorenzo; Butta', Paolo; Pisante, Adriano. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - STAMPA. - 224:2(2017), pp. 659-707. [10.1007/s00205-017-1086-3]

Stochastic Allen–Cahn approximation of the mean curvature flow: large deviations upper bound

BERTINI MALGARINI, Lorenzo;BUTTA', Paolo;PISANTE, Adriano
2017

Abstract

Consider the Allen–Cahn equation on the d-dimensional torus, d = 2, 3, in the sharp interface limit. As is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen–Cahn equation and describe its large deviation asymptotics in a joint sharp interface and small noise limit. Relying on previous results on the variational convergence of the action functional, we prove the large deviations upper bound. The corresponding rate function is finite only when there exists a time evolving interface of codimension one between the two stable phases. The zero level set of this rate function is given by the evolution by mean curvature in the sense of Brakke. Finally, the rate function can be written in terms of the sum of two non-negative quantities: the first measures how much the velocity of the interface deviates from its mean curvature, while the second is due to the possible occurrence of nucleation events.
2017
analysis; mathematics (miscellaneous); mechanical engineering
01 Pubblicazione su rivista::01a Articolo in rivista
Stochastic Allen–Cahn approximation of the mean curvature flow: large deviations upper bound / BERTINI MALGARINI, Lorenzo; Butta', Paolo; Pisante, Adriano. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - STAMPA. - 224:2(2017), pp. 659-707. [10.1007/s00205-017-1086-3]
File allegati a questo prodotto
File Dimensione Formato  
Bertini_Stochastic-Allen–Cahn_2017.pdf

solo gestori archivio

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 889.12 kB
Formato Adobe PDF
889.12 kB Adobe PDF   Contatta l'autore
Bertini_postprint_Stochastic-Allen–Cahn_2017.pdf

accesso aperto

Tipologia: Documento in Post-print (versione successiva alla peer review e accettata per la pubblicazione)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 540.71 kB
Formato Adobe PDF
540.71 kB Adobe PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/953467
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 9
  • ???jsp.display-item.citation.isi??? 10
social impact