Zhu, Qiang; Peirce, Anthony; Chadam, John Initiation of shape instabilities of free boundaries in planar Cauchy- Stefan problems. (English) Zbl 0816.35159 Eur. J. Appl. Math. 4, No. 4, 419-436 (1993). 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. Cited in 2 Documents MSC: 35R35 Free boundary problems for PDEs 80A22 Stefan problems, phase changes, etc. Keywords:instabilities; solidification; 1-phase Stefan problem with surface tension Citations:Zbl 0544.35088 PDFBibTeX XMLCite \textit{Q. Zhu} et al., Eur. J. Appl. Math. 4, No. 4, 419--436 (1993; Zbl 0816.35159) Full Text: DOI 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 [5] Bender, Advanced Mathematical Methods for Scientists and Engineers (1978) [6] Chadam, College de France Seminar VI pp 27– (1984) [7] Carslaw, Conduction of Heat in Solids (1959) [8] Rubinstein, IMA J. Appl. Math 28 (1982) [9] DOI: 10.1063/1.1692217 · Zbl 0182.29001 · doi:10.1063/1.1692217 [10] Dewynne, J. Austral. Math. Soc. 31 pp 81– (1989) [11] Ricci, Euro. J. Appl Math. 2 pp 1– (1991) [12] Ockendon, Proc. Seminar on Free Boundary Problems (1979) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.