Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations

We work in the setting of the whole plane (and possibly the whole three-dimensional space in the Gross-Pitaevskii case), in the asymptotic limit where \varepsilon , the characteristic lengthscale of the vortices, tends to 0, and in a situation where the number of vortices N_\varepsilon blows up as \varepsilon \to 0. The requirements are that N_\varepsilon should blow up faster than \vert\mathrm {log } \, \varepsilon \vert in the Gross-Pitaevskii case, and at most like \vert\mathrm {log } \, \varepsilon \vert in the parabolic case. Both results assume a well-prepared initial condition and regularity of the limiting initial data, and use the regularity of the solution to the limiting equations.