The supercooled Stefan problem in one dimension

Lincoln Chayes; Inwon C. Kim

doi:10.3934/cpaa.2012.11.845

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2012, Volume 11, Issue 2: 845-859. Doi: 10.3934/cpaa.2012.11.845

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The supercooled Stefan problem in one dimension

Lincoln Chayes^1, and
Inwon C. Kim^2,

1.
Dept. of Mathematics, UCLA, Los Angeles, CA 90095
2.
UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555

Received: November 30, 2009

Revised: July 31, 2011

Published: February 2012

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Abstract

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.

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Mathematics Subject Classification: 35Q79, 35Q82, 35K55.

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References

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