Hadžić, M; Raphaël, P; (2019) On melting and freezing for the 2D radial Stefan problem. Journal of the European Mathematical Society , 21 (11) pp. 3259-3341. 10.4171/jems/904.

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Abstract

We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball {r≤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)1/2e−2√2|ln(T−t)|√+O(1)(c+o(1))(T−t)k+12|ln(T−t)|k+12k, k∈N∗ as t→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∞−λ(t)∼⎧⎩⎨1logt1tk(logt)2, k∈N∗ as t→+∞ which correspond respectively to the fundamental stable freezing rate, and excited regimes which are codimension k stable

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