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

Sylvia Serfaty

Research output: Contribution to journalArticle

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 ε the characteristic lengthscale of the vortices, tends to 0, and in a situation where the number of vortices Nεblows up as ε → 0. The requirements are that Nεshould blow up faster than |log ε| in the Gross-Pitaevskii case, and at most like |log ε| 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 regim Nε≪|log ε| but not if Nεgrows faster.

Original languageEnglish (US)
Pages (from-to)713-768
Number of pages56
JournalJournal of the American Mathematical Society
Volume30
Issue number3
DOIs
StatePublished - 2017

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Limiting Equations
Mean-field Limit
Ginzburg-Landau Equation
Gross
Parabolic Equation
Vortex
Limiting
Regularity
Converge
Incompressible Euler Equations
Gross-Pitaevskii Equation
Asymptotic Limit
Vortex flow
Length Scale
Mean Field
Blow-up
Initial conditions
Euler equations
Tend
Three-dimensional

Keywords

  • Euler equation
  • Ginzburg-Landau
  • Gross-Pitaevskii
  • Hydrodynamic limit
  • Meanfield limit
  • Vortex liquids
  • Vortices

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations. / Serfaty, Sylvia.

In: Journal of the American Mathematical Society, Vol. 30, No. 3, 2017, p. 713-768.

Research output: Contribution to journalArticle

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