Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness
Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness
复制标题
DOI：
10.2140/pmp.2022.3.171
复制
发表时间：
2019-02
期刊：
影响因子：
--
通讯作者：
F. Delarue;S. Nadtochiy;Mykhaylo Shkolnikov
中科院分区：
文献类型：
--
作者：
F. Delarue;S. Nadtochiy;Mykhaylo Shkolnikov
我们考虑过冷斯特凡问题，该问题描述了过冷液体在一个空间维度上的冻结。即使在冷冻率急剧上升的情况下，对问题的概率重新表述也可以定义全局解决方案。我们通过将液体中的温度分布与冰生长过程的规律性联系起来，提供了此类解决方案的完整描述。后者被证明在 (i) 连续可微性、(ii) H 旧连续性和 (iii) 不连续性之间转换。特别是，在第二种情况下，我们重新发现了 Stefan 在 1889 年关于普通 Stefan 问题的开创性论文 [Ste89] 中指出的增长过程的平方根行为。在我们的第二个主要定理中，我们建立了全局解决方案的独特性，这是在存在爆炸时具有奇异自激励的增长过程背景下的第一个结果。
We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of the freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Holder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal paper [Ste89] from 1889 for the ordinary Stefan problem. In our second main theorem, we establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present.