Mean field limits for Ginzburg-Landau vortices

Serfaty, Sylvia

Séminaire Laurent Schwartz — EDP et applications (2015-2016), Exposé no. 3, 15 p.

We review results in the literature on asymptotic limits for the Ginzburg-Landau equations. We then present results where we show, by a modulated energy method, that 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 $\epsilon$, the characteristic lengthscale of the vortices, tends to $0$, and in a situation where the number of vortices $N_{\epsilon }$ blows up as $\epsilon \to 0$. The requirements are that $N_{\epsilon }$ should blow up faster than $|log\epsilon |$ in the Gross-Pitaevskii case, and at most like $|log\epsilon |$ 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.

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