On melting and freezing for the 2D radial Stefan problem

Mahir Hadžić; Pierre Raphaël

doi:10.4171/jems/904

JournalsjemsVol. 21, No. 11pp. 3259–3341

On melting and freezing for the 2D radial Stefan problem

Mahir Hadžić
King's College London, UK
Pierre Raphaël
Université de Nice Sophia Antipolis, Nice, France

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Abstract

We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball ${r \leq λ (t)}$ . We revisit the pioneering analysis of [31] and prove the existence in the radial class of finite time melting regimes

λ (t) = ⎩ ⎨ ⎧ (T - t)^{1/2} e^{- \frac{2}{2} ∣ l n (T - t) ∣ + O (1)} (c + o (1)) \frac{( T - t ) ^{\frac{k + 1}{2}}}{∣ l n ( T - t ) ∣ ^{\frac{k + 1}{2 k}}}, k \in N^{*} as t \to T

which respectively correspond to the fundamental stable melting rate, and a sequence of codimension $k$ excited regimes. Our analysis fully revisits a related construction for the harmonic heat flow in [60] by introducing a new and canonical functional framework for the study of type II (i.e. non-self-similar) blow up. We also show a deep duality between the construction of the melting regimes and the derivation of a discrete sequence of global-in-time freezing regimes

λ_{\infty} - λ (t) \sim {\frac{1}{l o g t} \frac{1}{t ^{k} ( l o g t ) ^{2}}, k \in N^{*} as t \to + \infty

which correspond respectively to the fundamental stable freezing rate, and excited regimes which are codimension $k$ stable.

Cite this article

Mahir Hadžić, Pierre Raphaël, On melting and freezing for the 2D radial Stefan problem. J. Eur. Math. Soc. 21 (2019), no. 11, pp. 3259–3341

DOI 10.4171/JEMS/904