Inserted: 12 feb 2007
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 $L^p$ ($p>1$) remain in $L^p$).