Gradient flow of the superconducting Ginzburg-Landau functional on the plane. (English) Zbl 0894.35107

Authors’ abstract: “First we prove the existence of global smooth solutions of the gradient flow of the superconducting Ginzburg-Landau (or Abelian-Higgs) functional on \(\mathbb{R}^2\). It is then proved that in the case of critical coupling, for a large class of initial data of arbitrary winding number \(N\), each solution converges in temporal gauge to a unique solution of the static equations of the same winding number. The proof has two essential ingredients. Firstly, a weighted energy identity is used to obtain spatial exponential decay of certain quantities uniformly in time. This implies the strong subsequently convergence to a static solution in the \(H^2\) 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 \(\omega\)-limit set of each solution is a point. The adiabatic approximation consists of finding, at each time, the \(L^2\)-closest point to the solution on the space of static solutions of the same winding number as the initial data. Special cases of the result imply that two vortices of opposite sign will annihilate for a large class of initial data and that a single vortex is asymptotically stable with respect to a large class of perturbations”.


35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
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