Abstract
In this note we summarize recent work on the global existence, finite-time blowup and asymptotic behavior of planar and spherical solidification solutions of one-phase Stefan problems with surface tension and kinetic undercooling. Special self-similar and travelling wave solutions motivate the results and turn out to be the global attractors of all high-symmetry solutions with the same desiderata. These results are used in determining the onset of shape instabilities in planar and spherical solidification.
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References
J. Ockendon, Linear and nonlinear stability of a class of moving free boundary problems, Proc. of Seminar on Free Boundary Problems, E. Magenese, ed., Rome: Inst. Naz. di Alta Mat. (1979).
J. Chadam and P. Ortoleva, The stabilizing effect of surface tension on the development of the free boundary in a planar, one dimensional, Cauchy-stefan problem, I. M. A. J. of Appl. Math., 30 (1983), pp. 57–66.
J. Chadam, S.D. Howison and P. Ortoleva, Existence and stability of spherical crystals growing in a supersaturated solution, I. M. A. J. of Appl. Math., 39 (1987), pp. 1–15.
H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Clarendon, Oxford (1959).
J. Chadam, J.-B. Baillon, M. Bertsch, P. Ortoleva and L.A. Peletier, Existence, uniqueness and asymptotic behavior of solutions of the planar, supersaturated solidification, Cauchy-Stefan problem, Lions-Brezis Seminar, Vol. VI, College de France, Res. Notes in Math., 109, Pitman, London (1984), pp. 27–47.
R. Ricci and W. Xie, On the stability of some solutions of the Stefan problem, European J. Appl. Math., 2 (1991), pp. 1–15.
A. Fasano and M. Primicerio, A critical case for the solvability of Stefan-like problems, Math. in Appl. Sci., 5 (1983), pp. 84–96.
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs (1964).
G. Caginalp and J. Chadam, Stability of interfaces with velocity correction term, Rocky Mtn. Math. J. 21 (1991), pp. 617–629.
R.J. Schaefer and M.E. Glicksman, Fully time-dependent theory for the growth of spherical crystal nuclei, J. Crystal Growth, 5 (1969), pp. 44–58.
N.N. Dewynne, S.D. Howison, J.R. Ockendon and W. Xie, Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary, J. Austral. Math. Soc., Ser. B, 31 (1989), pp. 81–96.
W.W. Mullins and R.F. Sekerka, Morphological stability of a particle growing by diffusion or heat flow, J. Appl. Phys. 34 (1964), pp. 323–329.
J.S. Langer, Instabilities and pattern formation in crystal growth, Rev. Mod. Phys., 52 (1980), pp. 1–28.
A. Meirmanov, Non-existence of classical solutions of a Stefan problem with surface tension on the free boundary, IMA preprint, to appear in this volume (1991).
J. Chadam and P. Ortoleva, On growth and form, Southern Illinois University at Carbondale (1980).
Q. Zhu, Z. Peirce and J. Chadam, Initiation of shape instabilities of free boundaries in planar Cauchy-Stefan problems, McMaster preprint, submitted for publication (1991).
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© 1993 Springer-Verlag New York, Inc.
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Chadam, J. (1993). Asymptotic Behavior of Solidification Solutions of Stefan Problems. In: Friedman, A., Spruck, J. (eds) Variational and Free Boundary Problems. The IMA Volumes in Mathematics and its Applications, vol 53. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8357-4_5
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DOI: https://doi.org/10.1007/978-1-4613-8357-4_5
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