2 research outputs found
Global attractors for Cahn-Hilliard equations with non constant mobility
We address, in a three-dimensional spatial setting, both the viscous and the
standard Cahn-Hilliard equation with a nonconstant mobility coefficient. As it
was shown in J.W. Barrett and J.W. Blowey, Math. Comp., 68 (1999), 487-517, one
cannot expect uniqueness of the solution to the related initial and boundary
value problems. Nevertheless, referring to J. Ball's theory of generalized
semiflows, we are able to prove existence of compact quasi-invariant global
attractors for the associated dynamical processes settled in the natural
"finite energy" space. A key point in the proof is a careful use of the energy
equality, combined with the derivation of a "local compactness" estimate for
systems with supercritical nonlinearities, which may have an independent
interest. Under growth restrictions on the configuration potential, we also
show existence of a compact global attractor for the semiflow generated by the
(weaker) solutions to the nonviscous equation characterized by a "finite
entropy" condition
Study of a system for the isothermal separation of components in a binary alloy with change of phase
An isothermal model describing the separation of the components of a
binary metallic alloy is considered. A process of phase transition is also
assumed to occur in the solder; hence, the state of the material is described
by two order parameters, i.e., the concentration c of the first component and
the phase field phi. A physical derivation is provided starting from energy
balance considerations. The resulting system of PDE\u27s consists of a rather
regular second order parabolic equation for phi coupled with a fourth order
relation of Cahn-Hilliard type for c with constraint and solution-dependent
mobility. Global existence of solutions is proved and several regularity
properties are discussed under more restrictive assumptions on the physical
parameters. Continuous dependence on data is shown in a special case. An
asymptotic analysis of the model is also performed, yielding at the limit step
a coupling of the original phase field equation with a Hele-Shaw like system
for c