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

We prove that in a certain asymptotic regime, solutions of the Gross-Pitaevskii equation converge to solutions of the incompressible Euler equation, and solutions to the parabolic Ginzburg-Landau equations converge to solutions of a limiting equation which we identify. 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 ε the characteristic lengthscale of the vortices, tends to 0, and in a situation where the number of vortices N_εblows up as ε → 0. The requirements are that N_εshould blow up faster than |log ε| in the Gross-Pitaevskii case, and at most like |log ε| 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. In the case of the parabolic Ginzburg-Landau equation, the limiting mean-field dynamical law that we identify coincides with the one proposed by Chapman-Rubinstein-Schatzman and E in the regim N_ε≪|log ε| but not if N_εgrows faster.