Research Article
Received 16 September 2011 Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.2505
MOS subject classification: 35K55; 35K59; 65M15; 80A22
A mathematical analysis of a nonisothermal
Allen–Cahn type system: error estimates
C. L. D. Vaz
a
and J. L. Boldrini
b
*
†
Communicated by H.-D. Alber
In this article, under certain conditions, we prove the regularity for the solutions of an Allen–Cahn phase-field type
system obtained as limits of approximate solutions constructed by using a semidiscrete spectral Galerkin method. With the
help of this improved regularity, as one compares to previous results, we then derive error estimates for the approximate
solutions in terms of the inverse of the eigenvalues of the Laplacian operator. The system under investigation may model
the evolution of solidification or melting of certain binary alloys. Copyright © 2012 John Wiley & Sons, Ltd.
Keywords: Allen–Cahn system; phase-field; phase transitions; binary alloy; solidification
1. Introduction
In this work, we analyze the following Allen–Cahn phase-field type system:
'
t
2
' D '.' 1/.1 2'/ jr'j .
1
c C
2
/ in Q, (1.1)
t
C `'
t
div.k.'/r/ D f .x, t/ in Q, (1.2)
c
t
div.D
1
.', /rc C D
2
.', /r'/ D 0 in Q, (1.3)
' D 0, D 0 c D 0 on S, (1.4)
'.x,0/ D '
0
.x/, .x,0/ D 0 c.x,0/ D c
0
.x/ in , (1.5)
where R
N
, N D 1 or 2, is an open and bounded domain with a C
2
boundary; T is a finite positive number; and Q D .0, T /
denotes the space–time cylinder with lateral surface S D @ .0, T /. This system may model the evolution of solidification or
melting processes occurring in certain binary alloys. In this case, the state of the alloy is characterized by the relative concentration
c (the proportion of solute in the solvent), the phase-field ', and the temperature . When ' D 0, the alloy is considered to be liquid;
' D 1, the alloy is solid. The region when 0 <'< 1 corresponds to the solid–liquid transition region, which is sometimes called the
mushy region.
The positive constants ,
1
,
2
, and ` are associated to material properties; f ./ is a given external field associated to the density
of heat sources or sinks; and k./ is associated to the thermal conductivity. D
1
./ and D
2
./ are diffusion coefficients of the solute in
the matrix of the solvent, that is, the other material constituting the binary alloy. '
0
./,
0
./, and c
0
./ are the initial conditions for the
phase-field, temperature, and solute concentration, respectively.
The previous phase-field equation (1.1) was basically derived by Beckermann et al. in [1], but here it is presented in a form obtained
before a final simplification was carried out. Thus, in a certain sense, Equation (3.9) can be considered more accurate than the final form
stated in [1]. The other two equations are obtained with rather general forms of balances of thermal energy and mass, and generalize
the cases of [1–3].
However, for preciser modeling, we must pay the price that the coupling nonlinearity in the phase-field equation (3.9), that is, the
term jr'j .
1
c C
2
/, involving products of the temperature and the concentration with derivatives of the phase-field, is much
more difficult to handle in mathematical terms than the usual classical coupling between the phase-field equation and the temperature
and concentration equations.
a
Matematica, Universidade Federal do Pará, Belem, Brazil
b
Matematica, Universidade Estadual de Campinas, Campinas, Brazil
*Correspondence to: J. L. Boldrini, Unicamp-IMECC, Rua Sérgio Buarque de Holanda 651, Campinas, SP, Brazil..
†
E-mail: boldrini@ime.unicamp.br
Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012