We study an evolution equation proposed by Chapman, Rubinstein, and 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" that decrease through the flow and allow us to get regularity results (solutions starting in Lp, p > 1, remain in Lp).
ASJC Scopus subject areas
- Applied Mathematics