Literature of phase boundaries

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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.

Further reading[edit]

  • G. Caginalp: Solidification problems as systems of nonlinear differential equations, Lectures in Applied Mathematics 23, 347–369 (1986)
  • I. Steinbach: Phase-field models in Materials Science – Topical Review, Modelling Simul. Mater. Sci. Eng. 17 (2009)073001


  • 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)