Stefan融化和冻结问题的微观模型

过冷Stefan问题：分形冻结与最大解的精细结构

The supercooled Stefan problem: fractal freezing and the fine structure of maximal solutions

本文研究了过冷Stefan问题，揭示了最大解的分形冻结特性及非通用性，并通过障碍问题理论分析了其正则性及稳定性。

聚合物钉扎动力学的标度极限和一维Stefan冻结问题

The scaling limit of polymer pinning dynamics and a one dimensional Stefan freezing problem

我们考虑了1+1维界面（或聚合物）在基底存在下的随机演化。这个随机过程是均匀钉扎模型的动力学版本。我们从一个远离平衡的配置开始：一个具有非平凡宏观高度分布的聚合物，并研究时空尺度界面的演化。在两种情况下，我们证明了这种重标度的界面在扩散尺度上具有标度极限（在两个维度上由$L$重新缩放的空间和由$L^2$重新缩放的时间，其中$L$表示接口的长度），我们描述：当与衬底的相互作用使得系统在平衡时未被钉扎时，然后通过求解Dirichlet边界条件下的热传导方程，给出了高度剖面的标度极限;当对基底的吸引力为无穷大时，标度极限是一个自由边界问题，它属于具有收缩边界的Stefan问题类，也被称为斯特凡冻结问题。此外，我们证明了该问题的解的存在性和正则性，直到一个最大的时间，在那里的边界碰撞。

Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness

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…

Zero kinetic undercooling limit in the supercooled Stefan problem

We study the solutions of the one-phase supercooled Stefan problem with kinetic undercooling, which describes the freezing of a supercooled liquid, in one spatial dimension. Assuming that the initial temperature lies between the equilibrium freezing point and the characteristic invariant temperature throughout the liquid our main theorem shows that, as the kinetic undercooling parameter tends to zero, the free boundary converges to the (possibly irregular) free boundary in the supercooled Stefan problem without kinetic undercooling, whose uniqueness has been recently established in [DNS19], [L…

Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem

The supercooled Stefan problem and its variants describe the freezing of a supercooled liquid in physics, as well as the large system limits of systemic risk models in finance and of integrate-and-fire models in neuroscience. Adopting the physics terminology, the supercooled Stefan problem is known to feature a finite-time blow-up of the freezing rate for a wide range of initial temperature distributions in the liquid. Such a blow-up can result in a discontinuity of the liquid-solid boundary. In this paper, we prove that the natural Euler time-stepping scheme applied to a probabilistic formula…

Stefan问题与最优布朗鞅输运的自由目标

The Stefan problem and free targets of optimal Brownian martingale transport

本文提出并解决自由目标最优布朗停时问题，其最优目标具有普适性，统一生成Stefan问题的不稳定（冻结）与稳定（融化）解，并首次获得不适定冻结问题的全局存在性与弱强唯一性。