A complete classification of singularities for radial solutions of the Stefan problem
Abstract
We investigate singularities of radial solutions of the classical two-phase and one-phase Stefan problem. In a remarkable result, M.A. Herrero and J.J.L. Velazquez show existence of a solution such that the free boundary described by g(t) satisfies in dimension n≥ 3 g(t)=cn √t |log t|(−1)/(n − 2) (1 + o(1)), t→ 0, and they conjecture that these asymptotics hold in general. Here we prove the conjecture for all radial solutions, which means complete asymptotic rigidity in that class. In order to obtain these precise asymptotics for the two-phase Stefan problem, for which frequency formulas and monotonicity formulas are unknown, we apply the Potential Reduction to ODE method introduced in arXiv:2608.18913 for the Neumann Bernoulli problem. The philosophy of that method is that the asymptotic shape of the free surface is governed by a precise ordinary differential equation. The fact that we can apply it in this paper to an equation of different type with quasilinear principal part, which is in our regime not the perturbation of a linear operator, confirms the versatility of the method.
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BibTeX
@misc{kriventsov2026complete,
title = {{A complete classification of singularities for radial solutions of the Stefan problem}},
author = {Dennis Kriventsov and Georg S. Weiss},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05434},
eprint = {2610.05434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05434},
}
RIS
TY - UNPB
TI - A complete classification of singularities for radial solutions of the Stefan problem
AU - Dennis Kriventsov
AU - Georg S. Weiss
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05434
UR - https://arxiv.org/abs/2610.05434
AB - We investigate singularities of radial solutions of the classical two-phase and one-phase Stefan problem. In a remarkable result, M.A. Herrero and J.J.L. Velazquez show existence of a solution such that the free boundary described by g(t) satisfies in dimension n≥ 3 g(t)=c_n √t |log t|^(−1)/(n − 2) (1 + o(1)), t→ 0, and they conjecture that these asymptotics hold in general. Here we prove the conjecture for all radial solutions, which means complete asymptotic rigidity in that class. In order to obtain these precise asymptotics for the two-phase Stefan problem, for which frequency formulas and monotonicity formulas are unknown, we apply the Potential Reduction to ODE method introduced in arXiv:2608.18913 for the Neumann Bernoulli problem. The philosophy of that method is that the asymptotic shape of the free surface is governed by a precise ordinary differential equation. The fact that we can apply it in this paper to an equation of different type with quasilinear principal part, which is in our regime not the perturbation of a linear operator, confirms the versatility of the method.
ER -