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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.
@article{AIHPC_2011__28_2_217_0,
author = {Ambrosio, Luigi and Mainini, Edoardo and Serfaty, Sylvia},
title = {Gradient flow of the {Chapman{\textendash}Rubinstein{\textendash}Schatzman} model for signed vortices},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {217--246},
publisher = {Elsevier},
volume = {28},
number = {2},
year = {2011},
doi = {10.1016/j.anihpc.2010.11.006},
mrnumber = {2784070},
zbl = {1233.49022},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.11.006/}
}
TY - JOUR AU - Ambrosio, Luigi AU - Mainini, Edoardo AU - Serfaty, Sylvia TI - Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices JO - Annales de l'I.H.P. Analyse non linéaire PY - 2011 SP - 217 EP - 246 VL - 28 IS - 2 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.11.006/ DO - 10.1016/j.anihpc.2010.11.006 LA - en ID - AIHPC_2011__28_2_217_0 ER -
%0 Journal Article %A Ambrosio, Luigi %A Mainini, Edoardo %A Serfaty, Sylvia %T Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices %J Annales de l'I.H.P. Analyse non linéaire %D 2011 %P 217-246 %V 28 %N 2 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.11.006/ %R 10.1016/j.anihpc.2010.11.006 %G en %F AIHPC_2011__28_2_217_0
Ambrosio, Luigi; Mainini, Edoardo; Serfaty, Sylvia. Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 217-246. doi: 10.1016/j.anihpc.2010.11.006
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