An analysis of a phase field model of a free boundary. (English) Zbl 0608.35080

The paper presents a new approach to solidification problems by assuming that the free boundary arising from a phase transition is not a sharp interface but has finite thickness. Assuming a free energy of Landau- Ginzburg from and relaxation-type dynamics, the author constructs a nonlinearly coupled parabolic system for temperature and phase function as mathematical model. The global existence of a unique solution is proved by constructing invariant regions for the system, and regularity results of Schauder type are obtained. An asymptotic analysis leads to the Gibbs-Thompson condition which relates the temperature at the interface to the surface tension and curvature.
Reviewer: J.Sprekels


35R35 Free boundary problems for PDEs
35K55 Nonlinear parabolic equations
Full Text: DOI


[1] L. I. Rubinstein, ”The Stefan Problem” · Zbl 0434.35086
[2] J. R. Ockendon & W. R. Hodgkins, eds., Moving Boundary Problems in Heat Flow and Diffusion, Oxford Univ. Press, Oxford (1975).
[3] D. G. Wilson, A. D. Solomon, & P. T. Boggs, eds., Moving Boundary Problems, Academic Press, New York (1978).
[4] A. Fasano & M. Primicerio, eds., Proce. Montecatini Symposium on Free and Moving Boundary Problems, Springer, Berlin Heidelberg New York (1981).
[5] C. M. Elliott & J. R. Ockendon, Weak and Variational Methods for Moving Boundary Problems, Pitman Publishing, London (1982). · Zbl 0476.35080
[6] A. Friedman, Variational Principles and Free-Boundary Problems, John Wiley and Sons, New York (1982). · Zbl 0564.49002
[7] O. A. Oleinik, ”A Method of Solution of the General Stefan Prob
[8] I.I. Kolodner, ”Free Boundary Problem for the Heat Equation with Applications to Problems of Change of Phase” · Zbl 0070.43803
[9] A. Friedman, ”The Stefan Problem in Several Space Variables”, · Zbl 0162.41903
[10] A. Friedman, ”Analyticity of the Free Boundary for the Stefan Problem”, Ar · Zbl 0329.35034
[11] D. Kinderlehrer & L. Nirenberg, ”The Smoothness of the Free Boundary in the One Phase Stefan Problem” · Zbl 0391.35060
[12] L. I. Rubinstein, ”The Stefan Problem: Comments on its Present State · Zbl 0434.35086
[13] L. A. Caffarelli, ”Continuity of the Temperature in the Stefan Problem” · Zbl 0406.35032
[14] L. A. Caffarelli, ”Some Aspects of the One Phase Stefan Problem”, Indiana Univ. Math. J, 27 (1980), 73–77. · Zbl 0393.35064
[15] L. A. Caffarelli & L. C. Evans, ”Continuity of the Temperature for the Two Phase Stefan Problem”, Arch. Rational Mech. Anal, (to appear). · Zbl 0516.35080
[16] J. Chad Am & P. Ortoleva, ”The Stabilizing Effect of Surface Tension on the Development of Free Boundaries”, Proc. Montecatini Symposium on Free and Moving Boundary Problems, Springer, Berlin Heidelberg New York (1981).
[17] B. Chalmers, Principles of Solidification, R. E. Krieger Publishing, Huntington, New York (1977).
[18] P. Hartman, Crystal Growth: An Introduction, North-Holland Publishing, Amsterdam (1973).
[19] J. W. Gibbs, Collected Works, Yale University Press, New Haven, (1948). · Zbl 0031.13504
[20] D. W. Hoffman & J. W. Cahn, ”A Vector Thermodynamics for Anisotropic Surfaces I. Fundamentals and Application to Plane Surface Junct
[21] J. W. Cahn & D. W. Hoffman, ”A Vector Thermodynamics for Anisotropic Surfaces II. Curved and Faceted Surfac
[22] W. W. Mullins, ”The Thermodynamics of Crystal Phases with Curved Interfaces: Special Case of Interface Isotropy and Hydrostatic Pressure”, Proc. Int. Conf. on Solid-Solid Phase Transformations, H. I. Erinson, et al., eds., TMS-AIME, Warrendale, Pennsylvania (1983).
[23] W. W. Mullins, ”Thermodynamic Equilibrium of a Crystal Sphere in a Fluid”, J. Chem. Phys. 81 (1984), 1436–1442.
[24] L. D. Landau & E. M. Lifshitz, Statistical Physics, Addison-Wesley Publishing, Reading, Massachusetts (1958). · Zbl 0080.19702
[25] C. J. Thompson, Mathematical Statistical Mechanics, MacMillan Co., New York (1972). · Zbl 0244.60082
[26] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford (1971).
[27] J. W. Cahn & J. E. Hilliard, ”Free Energy of a Nonuniform System I. Interfacial Free E
[28] J. W. Cahn & J. E. Hilliard, ”Free Energy of a Nonuniform System III. Nucleation in a Two Componente Incompressible
[29] J. S. Langer, ”Theory of the Condensation Poi
[30] P. C. Hohenberg & B. I. Halperin, ”Theory of Dynamic Critical Phenomena”, R
[31] G. Fix & J. T. Lin, Paper in preparation.
[32] W. W. Mullins & R. F. Sekerka, ”Morphological Stability of a Particle Growing by Diffusion or Heat F
[33] W. W. Mullins & R. F. Sekerka, ”Stability of a Planar Interface During Solidification of a Dilute Binary Al
[34] J. R. Ockendon, ”Linear and Non-linear Stability of a Class of Moving Boundary Problems”, Proc. Sem. Pavia 1979 Technoprint
[35] J. Smith, ”Shape Instabilities and Pattern Formation in Solidification: A New Method for Numerical Solution of the Moving Boundary Prob · Zbl 0463.65080
[36] J. Smoller, Shock Waves and Reaction –Diffusion Equations, Springer-Verlag, Berlin Heidelberg New York (1983). · Zbl 0508.35002
[37] H. Weinberger, ”Invariant Sets for Weakly Coupled Parabolic and Ellipti
[38] K. Chueh, C. Conley, & J. Smoller, ”Positively Invariant Regions for Systems of Nonlinear Diffusion Equations” · Zbl 0368.35040
[39] J. Bebernes, K. Chueh, & W. Fulks, ”Some Applications of Invariance for Parabolic Systems” · Zbl 0402.35056
[40] N. Alikakos, ”An Application of the Invariance Principle to Reaction-Diffusion Equa · Zbl 0386.34046
[41] H. Amann, ”Invariant Sets and Existence Theorems for Semilinear Parabolic and Elliptic Systems · Zbl 0387.35038
[42] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1964). · Zbl 0144.34903
[43] D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York (1977). · Zbl 0361.35003
[44] O. A. Ladyzenskaya, V. A. Solonnikov. & N. N. Uralceva, ”Linear and Quasilinear Equations of Parabolic Type”, Trans. of Math. Monographs 23, American Mathematical Society, Providence (1968).
[45] S. D. Eidelman, Parabolic Systems, North Holland Publishing, Amsterdam (1969).
[46] M. S. Berger & L. E. Fraenkel, ”On the Asymptotic Solution of a Nonlinear Dirichlet Pr
[47] P. Fife & W. M. Greenlee, ”Interior Transition Layers for Elliptic Boundary Value Problems with a Small Parameter · Zbl 0309.35035
[48] A. Van Harten, ”Nonlinear Singular Perturbation Problems: Proofs of Correctness of a Formal Approximation Based on a Contraction Principle in a Banach Space · Zbl 0393.34037
[49] F. A. Howes, ”Boundary-interior Layer Interactions in Nonlinear Singular Perturbation Theory · Zbl 0385.34010
[50] M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York (1977). · Zbl 0368.47001
[51] K. Yosida, Functional Analysis, Springer-Verlag, Berlin Heidelberg New York (1965). · Zbl 0126.11504
[52] N. Hicks, Notes on Differential Geometry, Van Nostrand, Princeton, New Jersey (1965). · Zbl 0132.15104
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