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Initiation of shape instabilities of free boundaries in planar Cauchy- Stefan problems. (English) Zbl 0816.35159

Summary: The linearized shape stability of melting and solidifying fronts with surface tension is discussed in this paper by using asymptotic analysis. We show that the melting problem is always linearly stable regardless of the presence of surface tension, and that the solidification problem is linearly unstable without surface tension, but with surface tension it is linearly stable for those modes whose wave numbers lie outside a certain finite interval determined by the perameters of the problem. We also show that if the perturbed initial data is zero in the vicinity of the front, but otherwise quite general, it does not affect the stability. The present results complement those in J. Chadam and P. Ortoleva [IMA J. Appl. Math. 30, 57-66 (1983; Zbl 0544.35088)] which are only valid asymptotically for large time or equivalently for slow-moving interfaces. The theoretical results are verified numerically.

MSC:

35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.

Citations:

Zbl 0544.35088
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References:

[1] DOI: 10.1093/imamat/30.1.57 · Zbl 0544.35088 · doi:10.1093/imamat/30.1.57
[2] DOI: 10.1093/imamat/25.1.1 · doi:10.1093/imamat/25.1.1
[3] DOI: 10.1103/RevModPhys.52.1 · doi:10.1103/RevModPhys.52.1
[4] DOI: 10.1063/1.1702607 · doi:10.1063/1.1702607
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