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March  2012, 11(2): 845-859. doi: 10.3934/cpaa.2012.11.845

The supercooled Stefan problem in one dimension

Lincoln Chayes 1, and  Inwon C. Kim 2,

1. 

Dept. of Mathematics, UCLA, Los Angeles, CA 90095, United States
2. 

UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, United States

Received  December 2009 Revised  August 2011 Published  October 2011

We study the 1D contracting Stefan problem with two moving boundaries that describes the freezing of a supercooled liquid. The problem is borderline ill--posed with a density in excess of unity indicative of the dividing line. We show that if the initial density, $\rho_0(x)$ does not exceed one and is not too close to one in the vicinity of the boundaries, then there is a unique solution for all times which is smooth for all positive times. Conversely if the initial density is too large, singularities may occur. Here the situation is more complex: the solution may suddenly freeze without any hope of continuation or may continue to evolve after a local instant freezing but, sometimes, with the loss of uniqueness.
Keywords: Stefan problem, supercooling, phase transition.
Mathematics Subject Classification: 35Q79, 35Q82, 35K55.
Citation: Lincoln Chayes, Inwon C. Kim. The supercooled Stefan problem in one dimension. Communications on Pure & Applied Analysis, 2012, 11 (2) : 845-859. doi: 10.3934/cpaa.2012.11.845
References:
[1]

I. Athanasopoulos, L. Caffarelli and S. Salsa, Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems, The Annals of Mathematics, 143 (1996), 413-434.  Google Scholar

[2]

E. Di Benedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc., 282 (1984), 183-204.  Google Scholar

[3]

L. Chayes and G. Swindle, Hydrodynamic limits for one dimensional particle systems with moving boundaries, Ann. Probab., 24 (1996), 559-598.  Google Scholar

[4]

L. Chayes and I. C. Kim, A two-sided contracting Stefan problem,, submitted for publication., ().   Google Scholar

[5]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Volume 19, AMS, 1998.  Google Scholar

[6]

A. Fasano and M. Primicerio, General free-boundary problems for the heat equation I, J. Math. Anal. Appl., 57 (1977), 694-723.  Google Scholar

[7]

A. Fasano, M. Primicerio, S. Howison and J. Ockendon, Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension, Quart. Appl. Math., 48 (1990), 153-168.  Google Scholar

[8]

I. G. Götz, M. Primicerio and J. J. L. Velázquez, Asymptotic behaviour $(t\to +0)$ of the interface for the critical case of undercooled Stefan problem, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 13 (2002), 143-148.  Google Scholar

[9]

I. G. Götz and B. Zaltzman, Some criteria for the disappearance of the mushy region in the Stefan problem, Quart. Appl. Math., 53 (1995), 657-671.  Google Scholar

[10]

M. A. Herrero and J. J. L. Velázquez, Singularity formation in the one dimensional supercooled Stefan problem, Eur. J. Appl. Math., 7 (1994), 115-150.  Google Scholar

[11]

M. A. Herrero and J. J. L. Velázqez, The birth of a cusp in the two-dimensional, undercooled Stefan problem, Quart. Appl. Math., 58 (2000), 473-494.  Google Scholar

[12]

H. Ishii, On a certain estimate of the free boundary in the Stefan problem, J. Differential Equations, 42 (1981), 106-115.  Google Scholar

[13]

A. M. Meirmanov, "The Stefan Problem," Translated from the Russian by Marek Niezgdka and Anna Crowley. With an appendix by the author and I. G. Gtz. de Gruyter Expositions in Mathematics, 3. Walter de Gruyter & Co., Berlin, 1992. x+245 pp. ISBN: 3-11-011479-8  Google Scholar

show all references

References:
[1]

I. Athanasopoulos, L. Caffarelli and S. Salsa, Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems, The Annals of Mathematics, 143 (1996), 413-434.  Google Scholar

[2]

E. Di Benedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc., 282 (1984), 183-204.  Google Scholar

[3]

L. Chayes and G. Swindle, Hydrodynamic limits for one dimensional particle systems with moving boundaries, Ann. Probab., 24 (1996), 559-598.  Google Scholar

[4]

L. Chayes and I. C. Kim, A two-sided contracting Stefan problem,, submitted for publication., ().   Google Scholar

[5]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Volume 19, AMS, 1998.  Google Scholar

[6]

A. Fasano and M. Primicerio, General free-boundary problems for the heat equation I, J. Math. Anal. Appl., 57 (1977), 694-723.  Google Scholar

[7]

A. Fasano, M. Primicerio, S. Howison and J. Ockendon, Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension, Quart. Appl. Math., 48 (1990), 153-168.  Google Scholar

[8]

I. G. Götz, M. Primicerio and J. J. L. Velázquez, Asymptotic behaviour $(t\to +0)$ of the interface for the critical case of undercooled Stefan problem, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 13 (2002), 143-148.  Google Scholar

[9]

I. G. Götz and B. Zaltzman, Some criteria for the disappearance of the mushy region in the Stefan problem, Quart. Appl. Math., 53 (1995), 657-671.  Google Scholar

[10]

M. A. Herrero and J. J. L. Velázquez, Singularity formation in the one dimensional supercooled Stefan problem, Eur. J. Appl. Math., 7 (1994), 115-150.  Google Scholar

[11]

M. A. Herrero and J. J. L. Velázqez, The birth of a cusp in the two-dimensional, undercooled Stefan problem, Quart. Appl. Math., 58 (2000), 473-494.  Google Scholar

[12]

H. Ishii, On a certain estimate of the free boundary in the Stefan problem, J. Differential Equations, 42 (1981), 106-115.  Google Scholar

[13]

A. M. Meirmanov, "The Stefan Problem," Translated from the Russian by Marek Niezgdka and Anna Crowley. With an appendix by the author and I. G. Gtz. de Gruyter Expositions in Mathematics, 3. Walter de Gruyter & Co., Berlin, 1992. x+245 pp. ISBN: 3-11-011479-8  Google Scholar

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