# MathSciDoc: An Archive for Mathematician ∫

#### Analysis of PDEsmathscidoc:1912.43854

Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 52, (10), 1189-1212, 1999.10
Abstract We study the GinzburgLandau equation on the plane with initial data being the product of n wellseparated+ 1 vortices and spatially decaying perturbations. If the separation distances are O ( 1), l, we prove that the n vortices do not move on the time scale O(^2),=o(1\over); instead, they move on the time scale O(^-21\over) according to the law j= xj W, W= l j log| xl xj|, xj=(j, j) 2, 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. John &amp; Wiley Sons, Inc.
@inproceedings{fh1999on,
title={On the dynamical law of the GinzburgLandau vortices on the plane},
author={FH Lin, and Jack Xin},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210223520374418},
booktitle={Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences},
volume={52},
number={10},
pages={1189-1212},
year={1999},
}

FH Lin, and Jack Xin. On the dynamical law of the GinzburgLandau vortices on the plane. 1999. Vol. 52. In Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences. pp.1189-1212. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210223520374418.