Literature of phase boundaries

The mathematical literature of phase boundaries has evolved since 1831 when Gabriel Lamé and Benoît Clapeyron [1] studied the freezing of the ground. This basic problem became known as the classical Stefan model [2] after its reformulation in 1889 in which both phases were considered.

References

G. Lamé, B. P. Clapeyron, Memoire sur la solidification par refroiddissement d'un globe solide, Ann. Chem. Physics, 47, 250–256 (1831)
J. Stefan, Über einige Probleme der Theorie der Warmeleitung, S.-B Wien Akad. Mat. Natur, 98, 173–484, (1889)
L. A. Caffarelli, "Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J. 28, 53–70, (1979)
A. M. Meirmanov, On a classical solution of the multidimensional Stefan problem for quasi-linear parabolic equations, Matematicheskii Sbornik, 112, 170–192, (1980)
J. W. Gibbs, Collected Works, Yale University Press, New Haven, (1948)
B. Chalmers, Principles of Solidification, John Wiley & Sons, Inc. (1964)
X. Chen, F. Reitich, Local existence and uniqueness of solution of the Stefan Problem, J. Math. Anal. Appl. 162, 350–362, (1992)
E. Radkevitch, The Gibbs–Thomson correction and conditions for the solutions of modified Stefan Problem, Sov. Math. Doklady, 43, 1, (1991)
S. Luckhaus, Solutions for the two-phase Stefan Problem with Gibbs–Thomson law for the melting temperature, Euro. J. Appl. Math, 1, 101–111, (1990)
J. Duchon, R. Robert, Evolution d'une interface par capillarite et diffusion de volume, Ann. Inst. Henri Poincare, Analyse non lineaire, 1, 361–378, (1984)
Y. G. Chen, Y. Giga, S. Goto, Uniqueness and existence of viscosity solution of generalized mean curvature equations, J. Diff. Geom 33, 749–786, (1991)
C. Evans, J. Spruck, Motion by mean curvature, J. Diff. Geom, 33, 635–681, (1991)
H. M. Soner, Motion of a set by the curvature of its boundary, J. Diff. Geom, 101, 313–372, (1993)
O. A. Oleinik, A method of solution of the general Stefan problem, Sov. Math. Dokl. 1, 1350–1354, (1960)
L. D. Landau and E. M. Lifshitz, Statistical Physics (Part 1) (Third Edition), Pergamon, New York, p.145. (1980)
P. C. Hohenberg, B. I. Halperin, Theory of dynamics in critical phenomena, Rev. Mod. Phys. 49, 435–480, (1977)
J. W. Cahn, J. H. Hilliard, Free energy of a non-uniform system I, Interfacial free energy, J. of Chem. Physics 28, 258–267, (1957)
S. M. Allen, J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metal. Mater. 27, 1084–1095, (1979)
J. Langer, Theory of condensation point, Annals of Physics, 41, 108–157, (1967)
G. Caginalp, The role of microscopic physics in the macroscopic behavior of a phase boundary, Annals of Physics, 172, 136–155, (1986)
G. Caginalp, P. Fife, Higher order phase field models and detailed anisotropy, Phys. Review B 34, 4940–4943 (1986)
G. Caginalp, A microscopic derivation of macroscopic sharp interface problems involving phase transitions, J. of Statistical Physics, 59, 869–884, (1990)
G. Caginalp, The limiting behavior of a free boundary in the phase field model, CMU Research Report, 82–5, (1982)
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92, 205, (1986)
G. Caginalp, Mathematical models of phase boundaries, in Material instabilities in continuum mechanics: Related mathematical problems, (Edited by J. M. Ball, ed.), Lecture at Heriot–Watt University (1985).
G. Caginalp, E. Socolovsky, Efficient computation of a sharp interface by spreading via phase field methods, Applied Math. Letters 2, 117–120 (1989)
X. Chen, G. Caginalp, C. Eck, A rapidly converging phase field model, Discrete and Continuous Dynamical Systems, 15, 1017–1034 (2006)
G. Caginalp, X. Chen, Convergence of the phase field model to its sharp interface limits, Euro. J. of Applied Mathematics, 9, 417–445, (1998)
H. M. Soner, Convergence of the phase field equation to the Mullins–Sekerka problem with kinetic undercooling, Arch. Rational. Mech. Anal. 131, 139–197, (1995)
B. Stoth, Convergence of Cahn–Hilliard equation to the Mullins–Sekerka problem in spherical symmetry, J. of Differential Equations, 125, 154–183, (1996)
X. Chen, The Hele–Shaw Problem as area-preserving curve shortening motions, Arch. Rat. Mech. Anal. 123, 117–151, (1993)
X. Chen, Spectrums of the Allen–Cahn, Cahn–Hilliard, and phase field equations for generic interfaces, Comm. Partial Differential Equations, 19, 13711–1395, (1994)
S. Gatti, M. Grasselli, V. Pata, Exponential attractors for a conserved phase-field system, Physica D, 189, 31–48, (2004)
M. Grasselli, V. Pata, Attractor for a conserved phase-field system with hyperbolic heat conduction, Mathematical Methods in the Applied Sciences, 27, 1917–1934, (2004)
A. Miranville, R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo Law, Nonlinear Analysis, 71, 2278–2290 (2009)
G. Schimperna, U. Stefanelli, A quasi-stationary phase field model with micro-movements, Applied Mathematics and Optimization 50, 67–86, (2004)
L. Cherfils, S. Gatti, A. Miranville, Existence of global solutions to the Caginalp phase field system with dynamic boundary conditions and singular potentials, J. Math. Anal. Appl. 343, (2008), 557–566
N. Kenmochi and K. Shirakawa, Stability for steady state patterns in phase field dynamics associated with total variation energies, Nonlinear Analysis, Theory, Methods and Applications, 53, 425–440 (2003)
C. G. Gal, M. Grasselli, On the asymptotic behavior of Caginalp systems with dynamic boundary conditions, Communications on Pure and Applied Analysis, 9, 689–710, (2009)
C. G. Gal, M. Grasselli, A. Miranville, Robust exponential attractors for singularly perturbed equation with dynamical boundary conditions, NoDEA Nonlinear Differential Equations and Applications, 15, 535–556, (2008)

Literature of phase boundaries

Further reading

References