4,264 research outputs found
Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system
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.Comment: Key words: Cahn-Hilliard equation, phase field model, well-posedness,
long-time behavio
A temperature-dependent phase segregation problem of the Allen-Cahn type
In this paper we prove a local-in-time existence theorem for an
initial-boundary value problem related to a model of temperature-dependent
phase segregation that generalizes the standard Allen-Cahn's model. The problem
is ruled by a system of two differential equations, one partial the other
ordinary, interpreted as balances, respectively, of microforces and of
microenergy, complemented by a transcendental condition on the three unknowns,
that are: the order parameter entering the standard A-C equation, the chemical
potential, and the absolute temperature. The results obtained by the authors in
a recent paper and dealing with the isothermal case serve as a starting point
for our existence proof, which relies on a fixed-point argument involving the
Tychonoff-Schauder theorem.Comment: Key words: Allen-Cahn equation; integrodifferential system;
temperature variable; local existence
Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system
In this paper we propose a time discretization of a system of two parabolic
equations describing diffusion-driven atom rearrangement in crystalline matter.
The equations express the balances of microforces and microenergy; the two
phase fields are the order parameter and the chemical potential. The initial
and boundary-value problem for the evolutionary system is known to be well
posed. Convergence of the discrete scheme to the solution of the continuous
problem is proved by a careful development of uniform estimates, by weak
compactness and a suitable treatment of nonlinearities. Moreover, for the
difference of discrete and continuous solutions we prove an error estimate of
order one with respect to the time step.Comment: Key words: Cahn-Hilliard equation, phase field model, time
discretization, convergence, error estimate
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