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Stefan problem: well-posedness and stability theories in presence and absence of surface tension

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If you have a question about this talk, please contact Prof. Clément Mouhot .

The Stefan problem is a well-known free boundary problem modeling phase transitions, melting/freezing phenomena, or nucleation. In the presence of surface tension, it serves as a micro-scale description of a phase transition, while in the absence thereof it acts as a macro-scale description. Mathematically, in the former case it has a flavor of a non-local curvature-driven flow, while in the latter case it changes its character into a non-linear system of parabolic-hyperbolic type, amenable to maximum principle techniques.

I will survey recent results on the well-posedness and stability theory, introducing a new unifying functional framework for the two problems. The first consequence is a rigorous vanishing surface tension limit. Moreover, I will show a global stability result in absence of surface tension, thereby explaining a hybrid methodology combining high-order energy methods and quantitative Hopf-type lemmas.

This talk is part of the Geometric Analysis & Partial Differential Equations seminar series.

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