Vanishing specific heat for the classical solutions of a multidimensional Stefan problem with kinetic condition
Authors:
Fahuai Yi and Jinduo Liu
Journal:
Quart. Appl. Math. 57 (1999), 661-672
MSC:
Primary 35R35; Secondary 35K05, 80A22
DOI:
https://doi.org/10.1090/qam/1724298
MathSciNet review:
MR1724298
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Abstract: In this paper we prove that the multidimensional Hele-Shaw problem with kinetic condition at the free boundary is the limit case of the Stefan problem with kinetic condition at the free boundary in the classical sense when the specific heat $\varepsilon$ goes to zero. The method is the use of a fixed point theorem; the key step is to construct a suitable function space in which we can get the existence and uniform estimates with respect to $\varepsilon > 0$ at the same time as for classical solutions of the multidimensional Stefan problem with kinetic condition at the free boundary. For the sake of simplicity, we only consider one-phase problems in three space dimensions, although the method used here is also applicable for two-phase problems and any space dimensions.
- A. Visintin, Stefan problem with a kinetic condition at the free boundary, Ann. Mat. Pura Appl. (4) 146 (1987), 97–122. MR 916689, DOI https://doi.org/10.1007/BF01762361
- Wei Qing Xie, The Stefan problem with a kinetic condition at the free boundary, SIAM J. Math. Anal. 21 (1990), no. 2, 362–373. MR 1038897, DOI https://doi.org/10.1137/0521020
- Avner Friedman and Bei Hu, A Stefan problem for multidimensional reaction-diffusion systems, SIAM J. Math. Anal. 27 (1996), no. 5, 1212–1234. MR 1402437, DOI https://doi.org/10.1137/S0036141094272848
- E. V. Radkevich and B. O. Ètonkulov, On the existence of a classical solution to the problem of the swelling of glassy polymers, Mat. Zametki 57 (1995), no. 6, 875–888, 959 (Russian, with Russian summary); English transl., Math. Notes 57 (1995), no. 5-6, 615–624. MR 1362232, DOI https://doi.org/10.1007/BF02304557
- Ei-ichi Hanzawa, Classical solutions of the Stefan problem, Tohoku Math. J. (2) 33 (1981), no. 3, 297–335. MR 633045, DOI https://doi.org/10.2748/tmj/1178229399
- Avner Friedman and Bei Hu, The Stefan problem with kinetic condition at the free boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 1, 87–111. MR 1183759
- Yu. E. Hohlov and M. Reissig, On classical solvability for the Hele-Shaw moving boundary problems with kinetic undercooling regularization, European J. Appl. Math. 6 (1995), no. 5, 421–439. Complex analysis and free boundary problems (St. Petersburg, 1994). MR 1363756, DOI https://doi.org/10.1017/S0956792500001960
J. F. Rodrigues (ed.), Free Boundary Problems News 4, 11–12, University of Lisboa, 1994
- Fahuai Yi, Asymptotic behaviour of the solutions of the supercooled Stefan problem, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 1, 181–190. MR 1433091, DOI https://doi.org/10.1017/S030821050002357X
- E. V. Radkevich, Conditions for the existence of a classical solution of a modified Stefan problem (the Gibbs-Thomson law), Mat. Sb. 183 (1992), no. 2, 77–101 (Russian); English transl., Russian Acad. Sci. Sb. Math. 75 (1993), no. 1, 221–246. MR 1166953, DOI https://doi.org/10.1070/SM1993v075n01ABEH003381
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
- Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR 1329547
A. Visintin, Stefan problem with a kinetic condition at the free boundary, Ann. Mat. Pura Appl. 146, 97–122 (1987)
W. Xie, The Stefan problem with a kinetic condition at the free boundary, SIAM J. Math. Anal. 21, 362–373 (1990)
A. Friedman and B. Hu, Stefan problem for multi-dimensional reaction diffusion systems, SIAM J. Math. Anal. 27, 1212–1234 (1996)
E. V. Radkevich and B. O. Étonkulov, On the existence of a classical solution to the problem of the swelling of glassy polymers, Math. Notes 57, 615–624 (1995)
E. Hanzawa, Classical solution of the Stefan problem, Tohoku Math. J. 33, 297–335 (1981)
A. Friedman and B. Hu, The Stefan problem with kinetic condition at the free boundary, Ann. Scuola Norm. Pisa 19, 87–111 (1992)
Yu. E. Hohlov and M. Reissig, On classical solvability for the Hele-Shaw moving boundary problems with kinetic undercooling regularization, European J. Appl. Math. 6, 421–439 (1995)
J. F. Rodrigues (ed.), Free Boundary Problems News 4, 11–12, University of Lisboa, 1994
F. Yi. Asymptotic behaviour of the solutions of the supercooled Stefan problem, Proceedings of the Royal Society of Edinburgh 127 A, 181–190 (1997)
E. V. Radkevich, On conditions for the existence of a classical solution of the modified Stefan problem, Russian Acad. Sci. Sb. Math. 75, 221–246 (1993)
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964
J. Simon, Compact sets in the space $L^{p}\left ( 0, T; B \right )$, Ann. Mat. Appl. 146, 65–96 (1987)
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995
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