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Erschienen in:
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2020 | OriginalPaper | Buchkapitel

3. Mathematical Model

verfasst von : Vladimir Danilov, Roman Gaydukov, Vadim Kretov

Erschienen in: Mathematical Modeling of Emission in Small-Size Cathode

Verlag: Springer Singapore

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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.

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Vorheriges Kapitel Physical Basis for Field Emission
Nächstes Kapitel Numerical Simulation and its Results
Fußnoten
1
For the rigorous asymptotic analysis of the phase field system, see in the next sections.
 
2
For more details about the problems considered here and references to the literature, see [19].
 
Literatur
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)MathSciNetMATHCrossRef
2.
Alexiades, V.: Mathematical Modeling of Melting and Freezing Processes. CRC Press (1992)
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)ADSMathSciNetMATHCrossRef
4.
Bossavit, A., Damlamian, A., Fremond, M. (Eds.): Free Boundary Problems: Applications and Theory. Pitman (1985)
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)CrossRef
6.
7.
8.
9.
Caginalp, G., Chadam, J.: Stability of interfaces with velocity correction term. Rocky Mount. J. Math. 21(2), 617–629 (1991)MathSciNetMATHCrossRef
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)MathSciNetMATHCrossRef
11.
Caginalp, G., Fife, P.C.: Elliptic problems involving phase boundaries satisfying a curvature condition. IMA J. Appl. Math. 38, 195–217 (1987)MathSciNetMATHCrossRef
12.
Caginalp, G., McLeod, B.: The interior transition layer for ordinary differential equations arising from solidification theory. Quart. Appl. Math. 44, 155–168 (1986)MathSciNetMATH
13.
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)MathSciNetMATHCrossRef
15.
Chalmers, B.: Principles of solidification. Wiley Series on the Science and Technology of Materials (Book 28). Wiley (1964)
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 715
17.
Crowley, A.B., Ockendon, J.R.: Modelling mushy regions. Appl. Sci. Res. 44, 1–7 (1987)CrossRef
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)CrossRef
19.
Danilov, V.G., Maslov, V.P., Volosov, K.A.: Mathematical Modelling of Heat and Mass Transfer Processes. Kluwer Academic Publication (1995)
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)MathSciNetMATH
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)ADSMathSciNetMATHCrossRef
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)MathSciNetMATHCrossRef
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)
24.
25.
Egorov, Y.V.: Linear Differential Equations of Principal Type. Springer (1986)
26.
Elliott, C.M., Ockendon, J.R.: Weak and Variational Methods for Free and Moving Boundary Problems. Pitman Publishing, Boston (1982)MATH
27.
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)ADSMathSciNetCrossRef
29.
Gelfand, I.M., Shilov, G.E.: Generalized Functions: Properties and Operations. Academic Press (1964)
30.
Gibbs, J.W.: The Collected Works. Yale University Press, New Haven (1948)MATH
31.
Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View, 2nd edn. Springer, NY (1987)MATHCrossRef
32.
Hoffmann, K.H., Sprekels, J. (Eds.): Free Boundary Problems: Theory and Applications. Longman Scientific and Technical (1990)
33.
Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Modern Phys. 49(3), 435–479 (1977)ADSCrossRef
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)MathSciNetCrossRef
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)ADSCrossRef
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)ADSMathSciNetCrossRef
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)
38.
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)
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)MathSciNetMATHCrossRef
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)MathSciNetMATHCrossRef
42.
Maslov, V.P., Omel’yanov, G.A.: Asymptotic soliton-form solutions of equations with small dispersion. Russian Math. Surv. 36, 73–149 (1981)ADSMATHCrossRef
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)MATHCrossRef
44.
Meirmanov, A.M.: An example of nonexistence of a classical solution of the Stefan problem. Soviet Math. Dokl. 23, 564–566 (1981)
45.
Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)MathSciNetMATHCrossRef
46.
Oleinik, O.A.: Discontinuous solution of non-linear differential equations. AMS Transl. Ser. 2(26), 95–172 (1963)MathSciNetMATH
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)MATHCrossRef
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)MathSciNetMATH
49.
Primicerio, M.: Mushy region in phase–change problem. Methoden und Verfahren der mathematischen Physik 25, pp. 251–269. Peter Lang, Frankfurt/Main (1983)
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)
51.
Radkevich, E.V.: About asymptotic solution of a phase-field system. Differ. Equat. 29(3), 487–500 (1993)
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)ADSMathSciNetMATHCrossRef
53.
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)MATHCrossRef
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)
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)MathSciNetMATHCrossRef
57.
Visintin, A.: Models of Phase Transitions. Birkhäuser (1996)
58.
Metadaten
Titel
Mathematical Model
verfasst von
Vladimir Danilov
Roman Gaydukov
Vadim Kretov
Copyright-Jahr
2020
Verlag
Springer Singapore
Buch
Mathematical Modeling of Emission in Small-Size Cathode

Print ISBN: 978-981-15-0194-4

Electronic ISBN: 978-981-15-0195-1

Copyright-Jahr: 2020

DOI
https://doi.org/10.1007/978-981-15-0195-1_3