A gradient flow approach to an evolution problem arising in superconductivity
نویسندگان
چکیده
We study an evolution equation proposed by Chapman-Rubinstein-Schatzman as a mean-field model for the evolution of the vortex-density in a superconductor. We treat the case of a bounded domain where vortices can exit or enter the domain. We show that the equation can be derived rigorously as the gradient-flow of some specific energy for the Riemannian structure induced by the Wasserstein distance on probability measures. This leads us to some existence and uniqueness results and energy-dissipation identities. We also exhibit some “entropies” which decrease through the flow and allow to get regularity results (solutions starting in Lp (p > 1) remain in Lp).
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