### Abstract

We study the Ginzburg-Landau equation on the plane with initial data being the product of n well-separated +1 vortices and spatially decaying perturbations. If the separation distances are O(ε^{-1}), ε ≪ 1, we prove that the n vortices do not move on the time scale O(ε^{-2}λ_{ε}), λ_{ε} = o(log 1/ε); instead, they move on the time scale O(e^{-2} log 1/ε) according to the law ẋ_{j} = -∇_{xj}W, W = -Σ_{l≠j}log|x_{l} - X_{j}|, x_{j} = (ξ_{j}, η_{j}) ∈ ℝ^{2}, the location of the j^{th} vortex. The main ingredients of our proof consist of estimating the large space behavior of solutions, a monotonicity inequality for the energy density of solutions, and energy comparisons. Combining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law.

Original language | English (US) |
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Pages (from-to) | 1189-1212 |

Number of pages | 24 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 52 |

Issue number | 10 |

State | Published - Oct 1999 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*52*(10), 1189-1212.

**On the dynamical law of the Ginzburg-Landau vortices on the plane.** / Lin, Fang-Hua; Xin, J. X.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 52, no. 10, pp. 1189-1212.

}

TY - JOUR

T1 - On the dynamical law of the Ginzburg-Landau vortices on the plane

AU - Lin, Fang-Hua

AU - Xin, J. X.

PY - 1999/10

Y1 - 1999/10

N2 - We study the Ginzburg-Landau equation on the plane with initial data being the product of n well-separated +1 vortices and spatially decaying perturbations. If the separation distances are O(ε-1), ε ≪ 1, we prove that the n vortices do not move on the time scale O(ε-2λε), λε = o(log 1/ε); instead, they move on the time scale O(e-2 log 1/ε) according to the law ẋj = -∇xjW, W = -Σl≠jlog|xl - Xj|, xj = (ξj, ηj) ∈ ℝ2, the location of the jth vortex. The main ingredients of our proof consist of estimating the large space behavior of solutions, a monotonicity inequality for the energy density of solutions, and energy comparisons. Combining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law.

AB - We study the Ginzburg-Landau equation on the plane with initial data being the product of n well-separated +1 vortices and spatially decaying perturbations. If the separation distances are O(ε-1), ε ≪ 1, we prove that the n vortices do not move on the time scale O(ε-2λε), λε = o(log 1/ε); instead, they move on the time scale O(e-2 log 1/ε) according to the law ẋj = -∇xjW, W = -Σl≠jlog|xl - Xj|, xj = (ξj, ηj) ∈ ℝ2, the location of the jth vortex. The main ingredients of our proof consist of estimating the large space behavior of solutions, a monotonicity inequality for the energy density of solutions, and energy comparisons. Combining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law.

UR - http://www.scopus.com/inward/record.url?scp=0033465801&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033465801&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033465801

VL - 52

SP - 1189

EP - 1212

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 10

ER -