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.
|Original language||English (US)|
|Number of pages||24|
|Journal||Communications on Pure and Applied Mathematics|
|State||Published - Oct 1999|
ASJC Scopus subject areas
- Applied Mathematics