We continue the study of Ambrosio and Serfaty (2008)  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)  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)  in this context, we show the entropy argument of Ambrosio and Serfaty (2008)  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.
|Original language||English (US)|
|Number of pages||30|
|Journal||Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire|
|State||Published - 2011|
ASJC Scopus subject areas
- Mathematical Physics
- Applied Mathematics