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Publication 03-CNA-07
A one-phase Stefan problem via Monge-Kantorovich theory
Abstract: The nonlinearity in the weak formulation of the generic one-phase Stefan problem has a ''flat portion'' which renders impossible obtaining the convergence of the enthalpy interpolants needed for passing to the limit as we move from the discrete problem (discretized as a gradient flow w.r.t. the Wasserstein distance) to the continuous problem. We do, however, succeed in employing Wasserstein distance techniques for proving this convergence and, ultimately, existence of weak solutions for a class of nonlinear problems among which lies the homographic approximation of the Stefan problem. The challenge here is to prove precompactness in space-time of the standard time-interpolants. Convergence of these weak solutions to the weak solution of the Stefan problem is obtained in parallel to the uniqueness of solution of the approximate problem by an interesting adaptation of a technique due to the authors of [7]. The last section shows how the existence of weak solutions for the problem at hand can be obtained by means of the Monge-Kantorovich mass transfer theory without approximating the nonlinearity. |
