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Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations


Author: Sylvia Serfaty
Journal: J. Amer. Math. Soc. 30 (2017), 713-768
MSC (2010): Primary 35Q56, 35K55, 35Q55, 35Q31, 35Q35
DOI: https://doi.org/10.1090/jams/872
Published electronically: October 18, 2016
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Abstract: 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 $ \vert\mathrm {log } \, \varepsilon \vert$ in the Gross-Pitaevskii case, and at most like $ \vert\mathrm {log } \, \varepsilon \vert$ 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 \vert\mathrm {log } \, \varepsilon \vert$, but not if $ N_\varepsilon $ grows faster.


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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
Email: serfaty@ann.jussieu.fr

DOI: https://doi.org/10.1090/jams/872
Keywords: Ginzburg-Landau, Gross-Pitaevskii, vortices, vortex liquids, mean-field limit, hydrodynamic limit, Euler equation
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
Article copyright: © Copyright 2016 American Mathematical Society