### 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 N_{ε}blows up as ε → 0. The requirements are that N_{ε}should 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 N_{ε}≪|log ε| but not if N_{ε}grows faster.

Original language | English (US) |
---|---|

Pages (from-to) | 713-768 |

Number of pages | 56 |

Journal | Journal of the American Mathematical Society |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - 2017 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the American Mathematical Society*,

*30*(3), 713-768. https://doi.org/10.1090/jams/872

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

Research output: Contribution to journal › Article

*Journal of the American Mathematical Society*, vol. 30, no. 3, pp. 713-768. https://doi.org/10.1090/jams/872

}

TY - JOUR

T1 - Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations

AU - Serfaty, Sylvia

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Euler equation

KW - Ginzburg-Landau

KW - Gross-Pitaevskii

KW - Hydrodynamic limit

KW - Meanfield limit

KW - Vortex liquids

KW - Vortices

UR - http://www.scopus.com/inward/record.url?scp=85017162634&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017162634&partnerID=8YFLogxK

U2 - 10.1090/jams/872

DO - 10.1090/jams/872

M3 - Article

AN - SCOPUS:85017162634

VL - 30

SP - 713

EP - 768

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 0894-0347

IS - 3

ER -