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Author: Sprekels, Jürgen × Type: Text × Type: report × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: long-time behavior × Subject: phase field model ×

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    Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Colli, Pierluigi; Gilardi, Geanni; Podio-Guidugli, Paolo; Sprekels, Jürgen
    We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the associated initial/boundary-value problem; moreover, we give a description of the relative $omega$-limit set.
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    Limiting problems for a nonstandard viscous Cahn-Hilliard system with dynamic boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    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 to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases.