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

Abstract

We study vortices for solutions of the perturbed Ginzburg-Landau equations Δu + (u/ε2)(1 -|u|2) = fε where fε is estimated in L2. We prove upper bounds for the Ginzburg-Landau energy in terms of ∥fε∥L2, and obtain lower bounds for ∥fε∥L2 in terms of the vortices when these form "unbalanced clusters" where σi di2 ≠ = (σid 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. © European Mathematical Society 2007.