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Mathematical Model

  • Vladimir DanilovEmail author
  • Roman Gaydukov
  • Vadim Kretov
Chapter
  • 193 Downloads
Part of the Heat and Mass Transfer book series (HMT)

Abstract

This chapter is a “mathematical” one. Here we collect the mathematical background related to the mathematical model of phase transition based on the phase field system introduced by G. Caginalp. Sections 3.1 and 3.2 of the chapter contain some preliminaries and considerations about mathematical models from the physical viewpoint. In Sect. 3.3, we give the results of asymptotic analysis applied to the phase field system. In Sect. 3.4, we discuss a new definition of the generalized solution to the phase field system which is stable under passing to the limiting Stefan–Gibbs–Thomson problem. Finally, in Sect. 3.5, we discuss an approach which is a combination of mathematical (asymptotic) investigation and numerical analysis.

References

  1. 1.
    Ablowitz, M.J., Zeppetella, A.: Explicit solutions of fisher’s equation for a special wave speed. Bull. Math. Biol. 41(6), 835–840 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alexiades, V.: Mathematical Modeling of Melting and Freezing Processes. CRC Press (1992)Google Scholar
  3. 3.
    Alikakos, N.D., Bates, P.W.: On the singular limit in a phase field model of phase transitions. Annales de l’institut Henri Poincaré (C) Analyse non linéaire 5(2), 141–178 (1988)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bossavit, A., Damlamian, A., Fremond, M. (Eds.): Free Boundary Problems: Applications and Theory. Pitman (1985)Google Scholar
  5. 5.
    Caginalp, G.: Surface tension and supercooling in solidification theory. In: Garrido, L. (ed.) Applications of Field Theory to Statistical Mechanics. Lecture Notes in Physics, vol. 216, pp. 216–226. Springer, Berlin, Heidelberg (1985)CrossRefGoogle Scholar
  6. 6.
    Caginalp, G.: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92(3), 205–245 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Caginalp, G.: The role of microscopic anisotropy in the macroscopic behavior of a phase boundary. Ann. Phys. 172, 136–155 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Caginalp, G.: Stefan and Hele-shaw type models as asymptotic limits of the phase field equations. Phys. Rev. A 39, 5887–5896 (1989)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Caginalp, G., Chadam, J.: Stability of interfaces with velocity correction term. Rocky Mount. J. Math. 21(2), 617–629 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Caginalp, G., Chen, X.: Convergence of the phase field model to its sharp interface limits. Eur. J. Appl. Math. 9(4), 417–445 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Caginalp, G., Fife, P.C.: Elliptic problems involving phase boundaries satisfying a curvature condition. IMA J. Appl. Math. 38, 195–217 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Caginalp, G., McLeod, B.: The interior transition layer for ordinary differential equations arising from solidification theory. Quart. Appl. Math. 44, 155–168 (1986)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cariello, F., Tabor, M.: Painleve expansions for nonintegrable evolution equations. Phys. D: Nonlinear Phenom. 39(1), 77–94 (1989)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chadam, J., Howison, S.D., Ortoleva, P.: Existence and stability for spherical crystals growing in a supersaturated solution. IMA J. Appl. Math. 39(1), 1–15 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Chalmers, B.: Principles of solidification. Wiley Series on the Science and Technology of Materials (Book 28). Wiley (1964)Google Scholar
  16. 16.
    Chen, X., Reitich, F.: Local existence and uniqueness of solutions of the stefan problem with surface tension and kinetic undercooling (November 1990). IMA Preprint Series 715Google Scholar
  17. 17.
    Crowley, A.B., Ockendon, J.R.: Modelling mushy regions. Appl. Sci. Res. 44, 1–7 (1987)CrossRefGoogle Scholar
  18. 18.
    Danilov, V.G.: On the relation between the Maslov-Whitham method and the weak asymptotics method. In: Kamiński, A., Oberguggenberger, M., Pilipović, S. (eds.) Linear and Non-Linear Theory of Generalized Functions and its Applications, vol. 88, pp. 55–65. Banach Center Publications, Warsaw (2010)CrossRefGoogle Scholar
  19. 19.
    Danilov, V.G., Maslov, V.P., Volosov, K.A.: Mathematical Modelling of Heat and Mass Transfer Processes. Kluwer Academic Publication (1995)Google Scholar
  20. 20.
    Danilov, V.G., Omel’yanov, G.A., Radkevich, E.V.: Asymptotic behavior of the solution of a phase field system, and a modified stefan problem. Differ. Equat. 31(3), 446–454 (1995)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Danilov, V.G., Omel’yanov, G.A., Radkevich, E.V.: Justification of asymptotics of solutions of the phase-field equations and a modified Stefan problem. Sbornik: Math. 186(12), 1753–1771 (1995)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Danilov, V.G., Omel’yanov, G.A., Radkevich, E.V.: Hugoniot-type conditions and weak solutions to the phase-field system. Eur. J. Appl. Math. 10, 55–77 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Danilov, V.G., Omel’yanov, G.A., Shelkovich, V.M.: Weak asymptotics method and interaction of nonlinear waves. Am. Math. Soc. Transl. 2, 208, pp. 33–163. Providence: American Mathematical Society (2003)Google Scholar
  24. 24.
    Danilov, V.G., Subochev, P.Y.: Wave solutions of semilinear parabolic equations. Theor. Math. Phys. 89, 1029–1046 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Egorov, Y.V.: Linear Differential Equations of Principal Type. Springer (1986)Google Scholar
  26. 26.
    Elliott, C.M., Ockendon, J.R.: Weak and Variational Methods for Free and Moving Boundary Problems. Pitman Publishing, Boston (1982)zbMATHGoogle Scholar
  27. 27.
    Fife, P.C., Gill, G.S.: The phase-field description of mushy zones. Phys. D: Nonlinear Phenom. 35, 267–275 (1989)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Fife, P.C., Gill, G.S.: Phase-transition mechanisms for the phase-field model under internal heating. Phys. Rev. A 43(2), 843–851 (1991)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Gelfand, I.M., Shilov, G.E.: Generalized Functions: Properties and Operations. Academic Press (1964)Google Scholar
  30. 30.
    Gibbs, J.W.: The Collected Works. Yale University Press, New Haven (1948)zbMATHGoogle Scholar
  31. 31.
    Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View, 2nd edn. Springer, NY (1987)zbMATHCrossRefGoogle Scholar
  32. 32.
    Hoffmann, K.H., Sprekels, J. (Eds.): Free Boundary Problems: Theory and Applications. Longman Scientific and Technical (1990)Google Scholar
  33. 33.
    Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Modern Phys. 49(3), 435–479 (1977)ADSCrossRefGoogle Scholar
  34. 34.
    Howison, S.D., Lacey, A.A., Ockendon, J.R.: Hele-shaw free-boundary problems with suction. Quar. J. Mech. Appl. Math. 41(2), 183–193 (1988)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Karma, A., Rappel, W.J.: Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys. Rev. E 53(4), R3017–R3020 (1996)ADSCrossRefGoogle Scholar
  36. 36.
    Kawahara, T., Tanaka, M.: Interactions of traveling fronts: An exact solution of a nonlinear diffusion equation. Phys. Lett. A 97(8), 311–314 (1983)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Kolmogorov, A.N., Petrovskii, N.G., Piskunov, N.S.: A study of the diffusion equation with increase in the quantity of matter, and its application to a biological problem. Bull. Moscow State Univ. Ser. A. Math. Mech. 1(6), 1–16 (1937). (in Russian)Google Scholar
  38. 38.
    Lacey, A.A., Tayler, A.B.: A mushy region in a Stefan problem. IMA J. Appl. Math. 30(3), 303–313 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Lashin, A.M.: An investigation of the dynamics of first-order phase transition during the directional solidification of a pure metal into an undercooled melt on the base of phase-field model (2001). (in Russian)Google Scholar
  40. 40.
    Luckhaus, S.: Solutions for the two-phase stefan problem with the Gibbs—Thomson law for the melting temperature. Eur. J. Appl. Math. 1(02), 101–111 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Luckhaus, S., Modica, L.: The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Ration. Mech. Anal. 107(1), 71–83 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Maslov, V.P., Omel’yanov, G.A.: Asymptotic soliton-form solutions of equations with small dispersion. Russian Math. Surv. 36, 73–149 (1981)ADSzbMATHCrossRefGoogle Scholar
  43. 43.
    Maslov, V.P., Tsupin, V.A.: Propagation of a shock wave in an isentropic gas with small viscosity. J. Soviet Math. 13, 163–185 (1980)zbMATHCrossRefGoogle Scholar
  44. 44.
    Meirmanov, A.M.: An example of nonexistence of a classical solution of the Stefan problem. Soviet Math. Dokl. 23, 564–566 (1981)Google Scholar
  45. 45.
    Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Oleinik, O.A.: Discontinuous solution of non-linear differential equations. AMS Transl. Ser. 2(26), 95–172 (1963)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Oleinik, O.A., Radkevich, E.V.: On the analyticity of solutions of linear partial differential equations. Math. USSR—Sbornik 19(4), 581–596 (1973)zbMATHCrossRefGoogle Scholar
  48. 48.
    Plotnikov, P.I., Starovoitov, V.N.: The Stefan problem with surface tension as a limit of the phase field model. Differ. Equat. 29(3), 395–404 (1993)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Primicerio, M.: Mushy region in phase–change problem. Methoden und Verfahren der mathematischen Physik 25, pp. 251–269. Peter Lang, Frankfurt/Main (1983)Google Scholar
  50. 50.
    Radkevich, E.V.: Gibbs-thomson amendment and conditions for the existence of a classical solution of the modified stefan problem. Dokl. Akad. Nauk 316(6), 1311–1315 (1991). (in Russian)Google Scholar
  51. 51.
    Radkevich, E.V.: About asymptotic solution of a phase-field system. Differ. Equat. 29(3), 487–500 (1993)Google Scholar
  52. 52.
    Radkevich, E.V.: On conditions for the existence of a classical solution of the modified Stefan problem (the Gibbs–Thomson law). Russian Academy Sci. Sbornik Mathematics 75, 221–246 (1993)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Radkevich, E.V.: On the heat Stefan wave. Dokl. Akad. Nauk USSR 47(1), 150–155 (1993)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Soner, H.M.: Influence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling. Arch. Ration. Mech. Anal. 131(2), 139–197 (1995)zbMATHCrossRefGoogle Scholar
  55. 55.
    Treves, J.F.: Introduction to Pseudodifferential and Fourier Integral Operators, vol. 1: Pseudodifferential Operators, second printing edn. University Series in Mathematics. Plenum Press, NY (1982)Google Scholar
  56. 56.
    Uchiyama, K.: The behavior of solutions of some non-linear diffusion equations for large time. J. Math. Kyoto Univ. 18(3), 453–508 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Visintin, A.: Models of Phase Transitions. Birkhäuser (1996)Google Scholar
  58. 58.
    Weiss, J., Tabor, M., Carnevale, G.: The painleve property for partial differential equations. J. Math. Phys. 24(3), 528–552 (1983)ADSMathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Vladimir Danilov
    • 1
    Email author
  • Roman Gaydukov
    • 1
  • Vadim Kretov
    • 1
  1. 1.National Research University Higher School of EconomicsMoscowRussia

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