Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations
HTML articles powered by AMS MathViewer

by Sylvia Serfaty
J. Amer. Math. Soc. 30 (2017), 713-768
Published electronically: October 18, 2016


We prove that in a certain asymptotic regime, solutions of the Gross-Pitaevskii equation converge to solutions of the incompressible Euler equation, and solutions to the parabolic Ginzburg-Landau equations converge to solutions of a limiting equation which we identify.

We work in the setting of the whole plane (and possibly the whole three-dimensional space in the Gross-Pitaevskii case), in the asymptotic limit where $\varepsilon$, the characteristic lengthscale of the vortices, tends to $0$, and in a situation where the number of vortices $N_\varepsilon$ blows up as $\varepsilon \to 0$. The requirements are that $N_\varepsilon$ should blow up faster than $|\mathrm {log } \varepsilon |$ in the Gross-Pitaevskii case, and at most like $|\mathrm {log } \varepsilon |$ in the parabolic case. Both results assume a well-prepared initial condition and regularity of the limiting initial data, and use the regularity of the solution to the limiting equations.

In the case of the parabolic Ginzburg-Landau equation, the limiting mean-field dynamical law that we identify coincides with the one proposed by Chapman-Rubinstein-Schatzman and E in the regime $N_\varepsilon \ll |\mathrm {log } \varepsilon |$, but not if $N_\varepsilon$ grows faster.

Similar Articles
Bibliographic Information
  • Sylvia Serfaty
  • Affiliation: Sorbonne Universités, UPMC Université Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France; and Institut Universitaire de France; and Courant Institute, New York University, 251 Mercer Street, New York, New York 10012
  • MR Author ID: 637763
  • Email:
  • Received by editor(s): July 21, 2015
  • Received by editor(s) in revised form: June 26, 2016, and July 2, 2016
  • Published electronically: October 18, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 30 (2017), 713-768
  • MSC (2010): Primary 35Q56, 35K55, 35Q55, 35Q31, 35Q35
  • DOI:
  • MathSciNet review: 3630086