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 $\Delta u+\frac{1}{{\epsilon }^2}u(1-|u{|}^2)=f_{\epsilon }$ where $f_{\epsilon }$ is estimated in $L^2$. We prove upper bounds for the Ginzburg–Landau energy in terms of $\Vert f_{\epsilon }{\Vert }_{L^2}$, and obtain lower bounds for $\Vert f_{\epsilon }{\Vert }_{L^2}$ in term of the vortices when these form “unbalanced clusters” where ${\sum }_i{d}_i^2\ne ({\sum }_i{d}_i{)}^2$.

These results will serve in Part II of this paper 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.