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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 $N$ blows up as $ε\to 0$. The requirements are that $N$ should 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 equation. 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 regime $N\ll |\log ε|$, but not if $N$ grows faster.