Book Review
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MathSciNet review:
1568189
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Book Information:
Author:
Morton E. \linebreak Gurtin
Title:
Thermomechanics of evolving phase boundaries in the plane
Additional book information:
Oxford University Press, Oxford and New York, 1993, xi+148 pp., US$54.00. ISBN 019-853694-1.
U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, 175–196. MR 845704
[2] F. J. Almgren, Jr., and L. Wang, Mathematical existence of crystal growth with Gibbs-Thomson curvature effects, Preprint, 1994.
[3] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. 1. Interfacial free energy, J. Chem. Phys. 28 (1958), 358-367.
S. J. Chapman, S. D. Howison, and J. R. Ockendon, Macroscopic models for superconductivity, SIAM Rev. 34 (1992), no. 4, 529–560. MR 1193011, DOI 10.1137/1034114
B. Derrida, J. L. Lebowitz, E. R. Speer, and H. Spohn, Dynamics of an anchored Toom interface, J. Phys. A 24 (1991), no. 20, 4805–4834. MR 1131256
R. Dobrushin, R. Kotecký, and S. Shlosman, Wulff construction, Translations of Mathematical Monographs, vol. 104, American Mathematical Society, Providence, RI, 1992. A global shape from local interaction; Translated from the Russian by the authors. MR 1181197, DOI 10.1090/mmono/104
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206
[8] G. J. Fix, Numerical simulation of free boundary problems using phase field methods, The Mathematics of Finite Elements and Applications IV (Uxbridge, 1981), Academic Press, London and New York, 1982, p. 265.
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR 906392
[11] W. Kurz and D. J. Fisher, Fundamentals of solidification, third ed., Trans Tech Publications, Aedermannsdorf, Switzerland, 1989.
Stephan Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature, European J. Appl. Math. 1 (1990), no. 2, 101–111. MR 1117346, DOI 10.1017/S0956792500000103
W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900–904. MR 78836
Oliver Penrose and Paul C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D 43 (1990), no. 1, 44–62. MR 1060043, DOI 10.1016/0167-2789(90)90015-H
James A. Sethian and John Strain, Crystal growth and dendritic solidification, J. Comput. Phys. 98 (1992), no. 2, 231–253. MR 1150905, DOI 10.1016/0021-9991(92)90140-T
[16] W. A. Tiller, The science of crystallization: Macroscopic phenomena and defect generation, Cambridge Univ. Press, Cambridge and New York, 1991.
[17] A. A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phase-field model for isothermal phase transitions in binary alloys, Phys. Rev. A 45(10) (1992), 7424-7439.
- [1]
- U. Abresch and J. S. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), 175-196. MR 845704 (88d:53001)
- [2]
- F. J. Almgren, Jr., and L. Wang, Mathematical existence of crystal growth with Gibbs-Thomson curvature effects, Preprint, 1994.
- [3]
- J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. 1. Interfacial free energy, J. Chem. Phys. 28 (1958), 358-367.
- [4]
- S. J. Chapman, S. D. Howison, and J. R. Ockendon, Macroscopic models for superconductivity, SIAM Rev. 34 (1992), 529-560. MR 1193011 (94b:82037)
- [5]
- B. Derrida, J. L. Lebowitz, E. R. Speer, and H. Spohn, Dynamics of an anchored Toom interface, J. Phys. A 24 (1991), 4805-4834. MR 1131256 (92g:82079)
- [6]
- R. Dobrushin, R. Kotecky, and S. Shlosman, Wulff construction: A global shape from local interaction, Transl. Math. Monographs., vol. 104, Amer. Math. Soc., Providence, RI, 1992. MR 1181197 (93k:82002)
- [7]
- L. C. Evans and J. Spruck, Motion of level sets by mean curvature. 1, J. Differential Geom. 33 (1991), 635-681. MR 1100206 (92h:35097)
- [8]
- G. J. Fix, Numerical simulation of free boundary problems using phase field methods, The Mathematics of Finite Elements and Applications IV (Uxbridge, 1981), Academic Press, London and New York, 1982, p. 265.
- [9]
- M. Gage and R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), 69-96. MR 840401 (87m:53003)
- [10]
- M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), 285-314. MR 906392 (89b:53005)
- [11]
- W. Kurz and D. J. Fisher, Fundamentals of solidification, third ed., Trans Tech Publications, Aedermannsdorf, Switzerland, 1989.
- [12]
- S. Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature, European J. Appl. Math. 1 (1990), 101-111. MR 1117346 (92i:80004)
- [13]
- W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900-904. MR 0078836 (17:1252g)
- [14]
- O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D 43 (1990), 44-62. MR 1060043 (91h:82034)
- [15]
- J. A. Sethian and J. Strain, Crystal growth and dendritic solidification, J. Comput. Phys. 98 (1992), 231-253. MR 1150905 (92k:80002)
- [16]
- W. A. Tiller, The science of crystallization: Macroscopic phenomena and defect generation, Cambridge Univ. Press, Cambridge and New York, 1991.
- [17]
- A. A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phase-field model for isothermal phase transitions in binary alloys, Phys. Rev. A 45(10) (1992), 7424-7439.
Review Information:
Reviewer:
John Strain
Journal:
Bull. Amer. Math. Soc.
32 (1995), 431-434
DOI:
https://doi.org/10.1090/S0273-0979-1995-00611-7
