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

Abstract

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 threedimensional 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 Nε blows up as ε → 0. The requirements are that Nε should blow up faster than |log ε| in the GrossPitaevskii 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 regime Nε ≪ |log ε|, but not if Nε grows faster.

Mean field limits for Ginzburg-Landau vortices

“…Here, for simplicity we omit a few assumptions dealing with the behavior at infinity, the full statements can be found in [Se3].…”

“…In order to treat a broader regime for N ε 1, we introduce in [Se3] an alternate method, based on a "modulated energy", which exploits the (assumed) regularity and stability of the solution to the limit equation. The method is robust and works for dissipative as well as conservative equations, as well as for variants with gauge [Se3] or with "pinning" weight [DS].…”

“…The method is robust and works for dissipative as well as conservative equations, as well as for variants with gauge [Se3] or with "pinning" weight [DS]. It avoids the question of understanding the detailed dynamics of the vortices (their distances, etc).…”

Mean-Field Limits for Some Riesz Interaction Gradient Flows

“…Very recently, in the context of the 2D Gross-Pitaevskii and parabolic Ginzburg-Landau equations, Serfaty [28] proposed a new way of proving such mean-field limits 1 , based on a Gronwall argument for the so-called modulated energy, which is some adapted measure of the distance to the (postulated) limit. This idea of proof originates in the relative entropy method first introduced by Yau [31] for hydrodynamic limits (see e.g.…”

“…Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty [28] in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open. …”