Title: The supercooled Stefan problem with transport noise: weak solutions and blow-up Show Chinese title

Title: 过冷Stefan问题与输运噪声：弱解和爆破

Abstract: We derive two weak formulations for the one-dimensional supercooled Stefan problem with transport noise on the quarter plane. The first captures a continuously evolving system, while the second resolves blow-ups by allowing for suitable jump discontinuities in the evolution of the temperature profile and the freezing front. For the first formulation, we establish a probabilistic representation in terms of a conditional McKean--Vlasov problem, and we show that there is finite time blow-up with positive probability as soon as part of the initial temperature profile exceeds a critical value. On the other hand, the system evolves continuously according to a unique solution when the initial profile is everywhere below this value. In the presence of blow-ups, we exploit the conditional McKean--Vlasov problem to find probabilistic solutions of the second weak formulation and we show that, among these, there is a particular solution of minimal temperature increase over time. Finally, we characterize the jump discontinuities of this solution as the minimal resolution of instabilities with respect to an infinitesimal external heat transfer. A general uniqueness result remains an open problem. Show Chinese abstract

Abstract: 我们为带有传输噪声的一维过冷Stefan问题在第一象限平面上推导出两种弱形式。 第一种形式描述了一个连续演化的系统，而第二种形式通过允许温度分布和凝固前沿演化中的适当跳跃不连续性来解决爆破现象。 对于第一种形式，我们建立了一个条件McKean--Vlasov问题的概率表示，并且我们证明只要初始温度分布的一部分超过临界值，就会在有限时间内以正概率发生爆破。 另一方面，当初始分布处处低于该值时，系统根据唯一解连续演化。 在存在爆破的情况下，我们利用条件McKean--Vlasov问题找到第二种弱形式的概率解，并且我们证明在这些解中，有一个解在时间上具有最小的温度增加。 最后，我们将该解的跳跃不连续性表征为相对于无限小外部热传递的不稳定性最小化解。 一个一般的唯一性结果仍然是一个开放问题。