Notes on the limit equation of vortex motion for the Ginzburg-Landau equation with Neumann condition

This paper deals with the motion law of vortices in the limit as epsilon -- > 0 of the Ginzburg-Landau equation u(t) = Deltau + (1/epsilon (2))(1 - \u\ (2))u, u = (u(1),u(2))(T), in a planar contractible domain with Neumann bou ndary condition, where the vortices are meant by zeros of a solution. As ep silon --> 0, applying the argument by Jerrard-Soner to the Neumann case yie lds an ordinary differential equation, called a limit equation, describing the dynamics of the vortices. We show that the limit equation can be writte n by using the Green function with Dirichlet condition and the Robin functi on of it. With this nice form we discuss the dynamics of a single or two vo rtices together with equilibrium states of the limit equation. In addition for. the disk domain an explicit form of the equation is proposed and the d ynamics for multi-vortices is investigated.