The two-phase Stefan problem describes the temperature distribution in ahomogeneous 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 whichis transported by the difference (or jump) of the normal derivatives of thetemperature in each phase. We establish local-in-time well-posedness and aglobal-in-time stability result for arbitrary sufficiently smooth domains andsmall initial temperatures. To this end, we develop a higher-order energy withnatural weights adapted to the problem and combine it with Hopf-typeinequalities. This extends the previous work by Hadzic and Shkoller [31,32] onthe one-phase Stefan problem to the setting of two-phase problems, andsimplifies the proof significantly.
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