Local Well-Posedness and Global Stability of the Two-Phase Stefan Problem
Mahir Hadžić, Gustavo Navarro, Steve Shkoller · SIAM Journal on Mathematical Analysis · 2017
The two-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain, composed of two regions separated by an a priori unknown moving boundary which is transported by the difference (or jump) of the normal derivatives of the temperature in each phase. We establish local-in-time well-posedness and a global-in-time stability result for arbitrary sufficiently smooth domains and small initial temperatures. To this end, we develop a higher-order energy with natural weights adapted to the problem and combine it with Hopf-type inequalities. This extends the previous work by Hadžić and Shkoller [ Comm. Pure Appl. Math., 68 (2015), pp. 689--757; Philos. Trans. A, 373 (2015), 20140284] on the one-phase Stefan problem to the setting of two-phase problems, and simplifies the proof significantly.