We develop a gradient-ﬂow framework based on the Wasserstein metric for a parabolic moving boundary problem that models crystal dissolution and precipitation. In doing so we derive a new weak formulation for this moving-boundary problem and we show that this formulation is well posed. In addition, we develop a new uniqueness technique based on the framework of gradient ﬂows with respect to the Wasserstein metric. With this uniqueness technique, the Wasserstein framework becomes a complete well-posedness setting for this parabolic moving-boundary problem.
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Jacobus W. Portegies, Mark A. Peletier, Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient ﬂows. Interfaces Free Bound. 12 (2010), no. 2, pp. 121–150DOI 10.4171/IFB/229