Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow Part I: Study of the perturbed Ginzburg-Landau equation

We study vortices for solutions of the perturbed Ginzburg-Landau equations Δu + (u/ε²)(1 -|u|²) = f_ε where f_ε is estimated in L². We prove upper bounds for the Ginzburg-Landau energy in terms of ∥f_ε∥L², and obtain lower bounds for ∥f_ε∥L₂ in terms of the vortices when these form "unbalanced clusters" where σ_i d_i² ≠ = (σ_id _i)². These results will serve in Part n of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg-Landau heat flow, which allow one to study various phenomena occurring in this flow, including vortex collisions; they allow in particular extending the dynamical law of vortices beyond collision times.