Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices

We continue the study of Ambrosio and Serfaty (2008) [4], on the Chapman–Rubinstein–Schatzman–E evolution model for superconductivity, viewed as a gradient flow on the space of measures equipped with the quadratic Wasserstein structure. In Ambrosio and Serfaty (2008) [4], we considered the case of positive (probability) measures, while here we consider general real measures, as in the physical model. Understanding the evolution as a gradient flow in this context gives rise to several new questions, in particular how to define a “Wasserstein” distance for signed measures. We generalize the minimizing movement scheme of Ambrosio et al. (2005) [3], in this context, we show the entropy argument of Ambrosio and Serfaty (2008) [4] still carries through, and derive an evolution equation for the measure which contains an error term compared to the Chapman–Rubinstein–Schatzman–E model. Moreover, we also show the same applies to a very similar dissipative model on the whole plane.