We establish vortex dynamics for the time-dependent Ginzburg–Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.
Kurzke, M., & Spirn, D. (2014). Vortex liquids and the Ginzburg-Landau equation. Forum of Mathematics, Sigma, 2, doi:10.1017/fms.2014.6