We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature. The former is subject to a dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a fixed point argument. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor as well as of an exponential attractor.

The non-isothermal Allen-Cahn equation with dynamic boundary conditions

GRASSELLI, MAURIZIO
2008-01-01

Abstract

We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature. The former is subject to a dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a fixed point argument. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor as well as of an exponential attractor.
2008
File in questo prodotto:
File Dimensione Formato  
GG-DCDS.pdf

Accesso riservato

: Post-Print (DRAFT o Author’s Accepted Manuscript-AAM)
Dimensione 356.33 kB
Formato Adobe PDF
356.33 kB Adobe PDF   Visualizza/Apri

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/11311/528713
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 73
  • ???jsp.display-item.citation.isi??? 72
social impact