We prove global well-posedness for the Gross-Pitaevskii equation on the plane for classes of initial data having nonzero topological degree at infinity and therefore infinite Ginzburg-Landau energy. These classes allow us to consider arbitrary configurations of vortices as initial data. Our work follows recent results of Patrick Gérard  and Clément Gallo , where the finite energy regime is treated.
"A remark on the Cauchy problem for the 2D Gross-Pitaevskii equation with nonzero degree at infinity." Differential Integral Equations 20 (3) 325 - 338, 2007. https://doi.org/10.57262/die/1356039505