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

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^-2λ_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 = -∇_xjW, W = -Σ_l≠jlog|x_l - X_j|, x_j = (ξ_j, η_j) ∈ ℝ², 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.