We consider a stochastic perturbation of the Allen-Cahn equation in a bounded interval [-a, b] with boundary conditions fixing the different phases at a and b. We investigate the asymptotic behavior of the front separating the two stable phases in the limit epsilon -> 0, when the intensity of the noise is root epsilon and a, b -> infinity with epsilon. In particular, we prove that it is possible to choose a = a(epsilon) such that in a suitable time scaling limit, the front evolves according to a one-dimensional diffusion process with a nonlinear drift accounting for a "soft" repulsion from a. We finally show that a "hard" repulsion can be obtained by an extra diffusive scaling.

Boundary effects on the interface dynamics for the stochastic Allen-Cahn equation / BERTINI MALGARINI, Lorenzo; S., Brassesco; Butta', Paolo. - STAMPA. - (2009), pp. 87-93. (Intervento presentato al convegno 15th International Congress on Mathematical Physics tenutosi a Rio de Janeiro; Brazil) [10.1007/978-90-481-2810-5_7].

Boundary effects on the interface dynamics for the stochastic Allen-Cahn equation

BERTINI MALGARINI, Lorenzo;BUTTA', Paolo
2009

Abstract

We consider a stochastic perturbation of the Allen-Cahn equation in a bounded interval [-a, b] with boundary conditions fixing the different phases at a and b. We investigate the asymptotic behavior of the front separating the two stable phases in the limit epsilon -> 0, when the intensity of the noise is root epsilon and a, b -> infinity with epsilon. In particular, we prove that it is possible to choose a = a(epsilon) such that in a suitable time scaling limit, the front evolves according to a one-dimensional diffusion process with a nonlinear drift accounting for a "soft" repulsion from a. We finally show that a "hard" repulsion can be obtained by an extra diffusive scaling.
2009
15th International Congress on Mathematical Physics
ginzburg-landau equation; phase field-equation; fluctuations; noise
04 Pubblicazione in atti di convegno::04b Atto di convegno in volume
Boundary effects on the interface dynamics for the stochastic Allen-Cahn equation / BERTINI MALGARINI, Lorenzo; S., Brassesco; Butta', Paolo. - STAMPA. - (2009), pp. 87-93. (Intervento presentato al convegno 15th International Congress on Mathematical Physics tenutosi a Rio de Janeiro; Brazil) [10.1007/978-90-481-2810-5_7].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/226872
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