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 Mat. Sb., 1992, Volume 183, Number 2, Pages 77–101 (Mi msb1468)

This article is cited in 31 scientific papers (total in 31 papers)

On conditions for the existence of a classical solution of the modified Stefan problem (the Gibbs–Thomson law)

E. V. Radkevich

Abstract: Sufficient conditions close to necessary are obtained for the existence of a solution (unique for $\sigma>0$ and $\beta>0$) of a solution of the modified Stefan problem on a small time interval.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1993, 75:1, 221–246

Bibliographic databases:

MSC: Primary 35K15, 80A22; Secondary 35R35
Received: 27.02.1991

Citation: E. V. Radkevich, “On conditions for the existence of a classical solution of the modified Stefan problem (the Gibbs–Thomson law)”, Mat. Sb., 183:2 (1992), 77–101; Russian Acad. Sci. Sb. Math., 75:1 (1993), 221–246

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Radkevich E., Eshonkulov B., “On the Existence of the Classical Solution of the Glassy Polymer Impregnation Problem”, Dokl. Akad. Nauk, 325:4 (1992), 668–673
2. E. V. Radkevich, “On the spectrum of the pencil in the Verigin–Muskat problem”, Russian Acad. Sci. Sb. Math., 80:1 (1995), 33–73
3. Radkevich E., “On the Asymptotic Solutions of a Phase Field System”, Differ. Equ., 29:3 (1993), 418–429
4. Radkevich E., “The Heat Stefan Wave”, Dokl. Akad. Nauk, 328:6 (1993), 657–661
5. E. V. Radkevich, B. O. Ètonkulov, “On the existence of classical solutions of the problem on swelling of glassy polymers”, Math. Notes, 57:6 (1995), 615–624
6. V. G. Danilov, G. A. Omel'yanov, E. V. Radkevich, “Justification of asymptotics of solutions of the phase-field equations and a modified Stefan problem”, Sb. Math., 186:12 (1995), 1753–1771
7. Danilov V., Omelyanov G., Radkevich E., “The Modified Stefan Problem as a Limit of Asymptotic Solution for Phase Field System”, Dokl. Akad. Nauk, 343:5 (1995), 586–589
8. Danilov V., Omelyanov G., Radkevich E., “Asymptotics of the Solution to the Phase Field System and the Modified Stefan Problem”, Differ. Equ., 31:3 (1995), 446–454
9. Fahuai Yi, “A note on the classical solution of a two-dimensional superconductor free boundary problem”, Eur J Appl Math, 7:2 (1996)
10. Yi F., “Classical Solution of Quasi-Stationary Stefan Problem”, Chin. Ann. Math. Ser. B, 17:2 (1996), 175–186
11. Chen X., Hong J., Yi F., “Existence, Uniqueness, and Regularity of Classical Solutions of the Mullins-Sekerka Problem”, Commun. Partial Differ. Equ., 21:11-12 (1996), 1705–1727
12. Tao Y., “Classical Solution of Verigin Problem with Surface Tension”, Chin. Ann. Math. Ser. B, 18:3 (1997), 393–404
13. Omelyanov G., Radkevich E., Danilov V., “The Problem of the Phase Transition in the Phase Field System”, Dokl. Akad. Nauk, 352:6 (1997), 731–734
14. Yi F., “Asymptotic Behaviour of the Solutions of the Supercooled Stefan Problem”, Proc. R. Soc. Edinb. Sect. A-Math., 127:Part 1 (1997), 181–190
15. Yi Fahuai, “A free boundary problem of type-I superconductivity”, Acta Mathematicae Applicatae Sinica English Series, 14:1 (1998), 48
16. Omel'yanov G. Danilov V. Radkevich E., “Asymptotic Solution of the Conserved Phase Field System in the Fast Relaxation Case”, Eur. J. Appl. Math., 9:Part 1 (1998), 1–21
17. Yoshiaki Kusaka, Atusi Tani, “On the Classical Solvability of the Stefan Problem in a Viscous Incompressible Fluid Flow”, SIAM J Math Anal, 30:3 (1999), 584
18. Yi F., Liu J., “Vanishing Specific Heat for the Classical Solutions of a Multidimensional Stefan Problem with Kinetic Condition”, Q. Appl. Math., 57:4 (1999), 661–672
19. Rodrigues J. Solonnikov V. Yi F., “On a Parabolic System with Time Derivative in the Boundary Conditions and Related Free Boundary Problems”, Math. Ann., 315:1 (1999), 61–95
20. Yi F., “On a Three-Dimensional Free Boundary Problem in Superconductivity Involving Mean Curvature”, Proc. R. Soc. Edinb. Sect. A-Math., 131:Part 1 (2001), 205–224
21. Christof Eck, Peter Knabner, Sergey Korotov, “A Two-Scale Method for the Computation of Solid–Liquid Phase Transitions with Dendritic Microstructure”, Journal of Computational Physics, 178:1 (2002), 58
22. Fahuai Yi, Yuqing Liu, “Two-Phase Stefan Problem as the Limit Case of Two-Phase Stefan Problem with Kinetic Condition”, Journal of Differential Equations, 183:1 (2002), 189
23. Schweizer B., “A Stable Time Discretization of the Stefan Problem with Surface Tension”, SIAM J. Numer. Anal., 40:3 (2002), 1184–1205
24. Fahuai Yi, “Global classical solution of Muskat free boundary problem”, Journal of Mathematical Analysis and Applications, 288:2 (2003), 442
25. Solonnikov V., “Lectures on Evolution Free Boundary Problems: Classical Solutions”, Mathematical Aspects of Evolving Interfaces, Lecture Notes in Mathematics, 1812, ed. Ambrosio L. Mimura M. Deckelnick K. Solonnikov V. Dziuk G. Soner H. Colli P. Rodrigues J., Springer-Verlag Berlin, 2003, 123–175
26. Eck C., “Homogenization of a Phase Field Model for Binary Mixtures”, Multiscale Model. Simul., 3:1 (2005), 1–27
27. Pruess J., Simonett G., “Stability of Equilibria for the Stefan Problem with Surface Tension”, SIAM J. Math. Anal., 40:2 (2008), 675–698
28. V. I. Voititskiy, N. D. Kopachevskiy, P. A. Starkov, “Multicomponent conjugation problems and auxiliary abstract boundary-value problems”, Journal of Mathematical Sciences, 170:2 (2010), 131–172
29. Kopachevsky N.D., Voytitsky V.I., “On the Modified Spectral Stefan Problem and its Abstract Generalizations”, Modern Analysis and Applications: Mark Krein Centenary Conference, Vol 2, Operator Theory Advances and Applications, 191, eds. Adamyan V., Berezansky Y., Gohberg I., Gorbachuk M., Gorbachuk V., Kochubei A., Langer H., Popov G., Birkhauser Verlag Ag, 2009, 381–394
30. Ch. Eck, B. Jadamba, P. Knabner, “Error Estimates for a Finite Element Discretization of a Phase Field Model for Mixtures”, SIAM J Numer Anal, 47:6 (2010), 4429
31. Pruess J. Simonett G. Zacher R., “Qualitative Behavior of Solutions for Thermodynamically Consistent Stefan Problems with Surface Tension”, Arch. Ration. Mech. Anal., 207:2 (2013), 611–667
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