Limiting problems for a nonstandard viscous Cahn-Hilliard system with dynamic boundary conditions
Pierluigi Colli, Gianni Gilardi, J\"urgen Sprekels
TL;DR
This paper analyzes a nonlinear phase-field diffusion system with dynamic boundary conditions, focusing on asymptotic limits and long-term behavior, including convergence as viscosity tends to zero and characterization of omega-limit sets.
Contribution
It introduces a dynamic boundary condition involving the Laplace-Beltrami operator and studies the asymptotic and long-term behavior of the system, extending previous models.
Findings
Solutions converge to the limit problem as viscosity approaches zero
The long-time behavior and omega-limit sets are characterized for both positive and zero viscosity
The dynamic boundary condition models additional surface phase transitions
Abstract
This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions…
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document
Limiting problems
for a nonstandard viscous Cahn–Hilliard system
with dynamic boundary conditions
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centerPierluigi Colli*(1)*
e-mail: [email protected]
Gianni Gilardi*(1)*
e-mail: [email protected]
Jürgen Sprekels*(2)*
e-mail: [email protected]
(1) Dipartimento di Matematica “F. Casorati”, Università di Pavia
via Ferrata 5, 27100 Pavia, Italy
(2) Department of Mathematics
Humboldt-Universität zu Berlin
Unter den Linden 6, 10099 Berlin, Germany
and
Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39, 10117 Berlin, Germany