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A one-dimensional, one-phase Stefan problem as gradient flow
Abstract: Under reasonable assumptions on the initial data, we expect the classical solution for the one-phase Stefan problem with no boundary heat exchange to behave in a certain physical way that is relatively easy to predict and prove. As in many situations, however, first we only prove existence of weak solutions for which means of proving similar properties are ostensibly more difficult due to the lack of regularity. In the present work we treat the evolution as a gradient flow with respect to the Wasserstein distance on a special manifold and employ techniques from the Monge-Kantorovich mass transfer theory to show that the weak solution at almost every time level preserves certain features of the initial data. Numerical simulations support our theoretical results.
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