Dynamics of a free boundary in a binary medium with variable thermal conductivity. (English. Russian original) Zbl 0944.35114

Omel’yanov, G. A.; Trushkov, V. V.

Math. Notes 66, No. 2, 181-189 (1999); translation from Mat. Zametki 66, No. 2, 231-241 (1999).

Summary: We construct an asymptotic solution of the phase field system with variable thermal conductivity different in domains occupied by different phases. We show that, depending on relations between parameters characterizing the substance, the dynamics of the free interface between the phases is determined by solutions of the classical or modified Stefan problems.

MSC:

35R35	Free boundary problems for PDEs
80A22	Stefan problems, phase changes, etc.

Keywords:

phase transition; Stefan problem; asymptotic solution; phase field system

References:

[1]	G. Caginalp, ”An analysis of a phase field model of a free boundary,”Arch. Rational Mech. Anal.,92, 205–245 (1986). · Zbl 0608.35080 · doi:10.1007/BF00254827
[2]	G. Caginalp and X. Chen, ”Phase field equations in the singular limit of sharp interface problem,” in:On the Evolution of Phase Boundaries (M. Gurtin and G. B. McFadden, editors), Vol. 43, IMA Vol. Math. Appl, Springer, New York (1992), pp. 1–28. · Zbl 0760.76094
[3]	L. Modica, ”The gradient theory of phase translations and the minimal interphase criterion,”Arch. Rational Mech. Anal.,98, 123–142 (1986). · Zbl 0616.76004
[4]	S. Luckhaus and L. Modica, ”The Gibbs-Thomson relation within the gradient theory of phase translations,”Arch. Rational Mech. Anal.,107, 71–83 (1989). · Zbl 0681.49012 · doi:10.1007/BF00251427
[5]	S. Luckhaus, ”Solutions of the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature,”European J. Appl. Math.,1, 101–111 (1990). · Zbl 0734.35159 · doi:10.1017/S0956792500000103
[6]	P. I. Plotnikov and V. N. Starovoitov, ”The Stefan problem as a limit of the phase field system,”Differentsial’nye Uravneniya [Differential Equations],29, No. 3, 461–471 (1993).
[7]	H. M. Soner, ”Convergence of the phase field equations to the Mullins-Sekerka problem with kinetic undercooling,”Arch. Rational Mech. Anal.,131, 139–197 (1995). · Zbl 0829.73010 · doi:10.1007/BF00386194
[8]	G. Caginalp, ”Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations,”Phys. Rev. A,39, 101–111 (1990). · Zbl 0712.35114
[9]	G. Caginalp and X. Chen, ”Convergence of the phase field model to its sharp interphase limits,”European J. Appl. Math.,9, No. 4, 417–445 (1998). · Zbl 0930.35024 · doi:10.1017/S0956792598003520
[10]	N. Alikakos and P. Bates, ”On the singular limit in a phase field model of a phase translation,”Ann. Inst. H. Poincaré. Phys. Théor.,5, 1–38 (1988). · Zbl 0696.35060
[11]	J. Carr and R. L. Pego, ”Metastable patterns in solutions of \(u_t = \varepsilon ^2 u_{xx} - f(u)\) ,”Comm. Pure Appl. Math.,42, 523–576 (1989). · Zbl 0685.35054 · doi:10.1002/cpa.3160420502
[12]	X. Chen and C. M. Elliott, ”Asymptotics for a parabolic double obstacle problem,”Proc. Roy. Soc. London. Ser. A,444, 429–445 (1994). · Zbl 0814.35044 · doi:10.1098/rspa.1994.0030
[13]	R. Nochetto and C. Verdi, ”Convergence of double obstacle problem to the generalized geometric motion of fronts,”SIAM J. Math. Anal.,26, No. 4, 1514–1526 (1995). · Zbl 0839.35008 · doi:10.1137/S0036141093255429
[14]	V. G. Danilov, G. A. Omel’yanov, and E. V. Radkevich, ”Asymptotics of the phase field system and the modified Stefan problem,”Differentsial’nye Uravneniya [Differential Equations],31, No. 3, 483–491 (1995). · Zbl 0855.35134
[15]	V. G. Danilov, G. A. Omel’yanov, and E. V. Radkevich, ”Justification of the asymptotics of the phase field system and the modified Stefan problem,”Mat. Sb. [Russian Acad. Sci. Sb. Math.],186, No. 12, 63–80 (1995). · Zbl 0870.35127
[16]	G. A. Omel’yanov and V. V. Trushkov, ”A geometric correction to the free boundary problem,”Mat. Zametki [Math. Notes],63, No. 1, 151–153 (1998). · Zbl 0922.35191
[17]	V. P. Maslov and G. A. Omel’yanov, ”Asymptotic soliton type solutions of equations with small dispersion,”Uspekhi Mat. Nauk [Russian Math. Surveys],36, No. 3, 63–126 (1981). · Zbl 0463.35073
[18]	V. P. Maslov, V. G. Danilov, and K. A. Volosov,Mathematical Modeling of Heat and Mass Transfer Processes [in Russian], Nauka, Moscow (1987). · Zbl 0645.73049