Convergence of the self-dual Ginzburg-Landau gradient flow

We prove convergence of the gradient flow of the Ginzburg-Landau energy functional on a Riemann surface in the self-dual Bogomolny case, in Coulomb gauge. The proof is direct and makes use of the associated nonlinear first order differential operators (the Bogomolny operators). One aim is to illustrate that the Bogomolny structure, which is known to be of great utility in the static elliptic case, can also be used effectively in evolution problems. We also identify the minimizers and minimum value of the energy when the Bogomolny bound is not achieved (below the Bradlow limit).