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

Abstract

We study vortices for solutions of the perturbed Ginzburg-Landau equations $\Delta u+ \frac{1}{\ep2} u(1-|u|^2)=\a$ where $\a$ is estimated in $L2$. We prove upper bounds for the Ginzburg-Landau energy in terms of $\|\a\|_{L2}$, and obtain lower bounds for $\|\a\|_{L2}$ in term of the vortices when these form ``unbalanced clusters" where \sum_i d_i2\neq \(\sum_i d_i\)2. These results will serve in Part II of this paper \cite{part2} to provide estimates on the energy-dissipation rates for solutions of the Ginzburg-Landau heat-flow, which allow to study various phenomena occurring in this flow, among which vortex-collisions; allowing in particular to extend the dynamical law of vortices passed collisions.