2016/07/03 by Mahir Hadžić, Hadzic, Mahir, Gustavo Navarro +3
Computer Science · Mathematics · #35B40 #35B65 #35M30 #35Q79 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1607.00681
openalex publication_date 2016/07/03 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
The two-phase Stefan problem describes the temperature distribution in a\nhomogeneous medium undergoing a phase transition such as ice melting to water.\nThis is accomplished by solving the heat equation on a time-dependent domain,\ncomposed of two regions separated by an a priori unknown moving boundary which\nis transported by the difference (or jump) of the normal derivatives of the\ntemperature in each phase. We establish local-in-time well-posedness and a\nglobal-in-time stability result for arbitrary sufficiently smooth domains and\nsmall initial temperatures. To this end, we develop a higher-order energy with\nnatural weights adapted to the problem and combine it with Hopf-type\ninequalities. This extends the previous work by Hadzic and Shkoller [31,32] on\nthe one-phase Stefan problem to the setting of two-phase problems, and\nsimplifies the proof significantly.\n