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

Abstract:

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 $\varepsilon$, 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 $|\mathrm {log } \varepsilon |$ in the Gross-Pitaevskii case, and at most like $|\mathrm {log } \varepsilon |$ 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 regime $N_\varepsilon \ll |\mathrm {log } \varepsilon |$, but not if $N_\varepsilon$ grows faster.