On the Dynamical Law of the Ginzburg-Landau Vortices on the Plane F.-H. LIN Courant Institute AND J. X. XIN University of Arizona 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(ε −2 log 1 ε ) according to the law ˙ x j = −∇ x j W , W = − ∑ 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. Com- bining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law. c  1999 John Wiley & Sons, Inc. 1 Introduction We consider the Ginzburg-Landau (G-L) equation, u t = ∆ x u +(1 −|u| 2 )u , x ∈ R 2 , (1.1) where u = u(t , x) is a complex-valued function defined for each t > 0 and x = (ξ, η) ∈ R 2 ; ∆ = ∂ ξξ + ∂ ηη denotes the two-dimensional Laplacian. The G-L equa- tion (1.1) admits vortex solutions of the form, Ψ n (x)= U n (r)e inθ , n = ±1, ±2,..., U n (0)= 0 , U n (+∞)= 1 , (1.2) where (r , θ) denote the polar coordinates on R 2 . The functions Ψ n (x) define com- plex planar vector fields, whose zeros are called vortices or defects. Among them, only the degree 1 vortices are dynamically stable; see Weinstein and Xin [10] for the whole-plane case, and Mironescu [7] and Lieb and Loss [3] for the related bounded domain case. Hence it makes sense to inquire about the motion law of the degree 1 vortices. Hereafter, we shall be concerned with degree +1 vortices, and use U to denote the profile of such a vortex. The G-L equation (1.1) defines a continuous-in-time deformation of the com- plex vector field u(·, x). So if the initial total winding number or degree at infinity is different from zero, one expects the dynamics to be organized around the motion of the zeros of u(t , x). A description of the dynamics of an ensemble of spatially separated vortices is a fundamental problem. The systematic formal asymptotic Communications on Pure and Applied Mathematics, Vol. LII, 1189–1212 (1999) c  1999 John Wiley & Sons, Inc. CCC 0010–3640/99/101189-24