Gradient Flow of the Superconducting Ginzburg-Landau Functional on the Plane

摘要

First we prove the existence of global smooth solutions of the gradient flow ofthe super-conducting Ginzburg Landau (or Abelian Higgs) functional on R2. Ti is then proved that in the case of critical coupling, for a large class if initial data of arbitrary winding number N, each solution converges in temporal gauge to a unique solution ingredients. Firstly a weighted energy identity is used to obtain spatial exponential decay of certain quantities uniformly in time. This implies the strong subsequential convergence to a static solution in the H2 norm. Secondly, an adiabatic approximation in the neighbourhood of the static solution space is used to prove that the solution converges without passing to subsequences. Thus the w-limit set of each solution is a point. The adiabatic approximation consists of finding, at each time, the L2-closest point to the solution on the space of static solutions of the same winding number as the initial data.