Existence Of Vortices In The Abelian Chern-Simons Model

In this paper,we study the existence of the doubly periodic vortices in the Abelian Chern-Simons model by means of heat-flow method.Firstly,we establish the existence of a global solution of the governing semilinear parabolic equation.Then we prove the convergence of the global solution to an equilibrium(i.e.,a static doubly periodic vortex solution to the Abelian Chern-Simons model)as time goes to infinity.Therefore,we get the existence of the doubly periodic vortices for the Abelian Chern-Simons model.Moreover,we provide an estimate on the convergence rate by using the Lojasiewicz Simon inequality.Finally,we present a few numerical examples which demonstrate the effectiveness of our convergence results.