Vortex Liquids and the Ginzburg–landau Equation

Abstract

We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

Hydrodynamic Limit of the Gross-Pitaevskii Equation

“…The function φ ε appearing in the statement of the theorem can be taken to be the minimizer of the energy (7) ψ → Remark 1. Results of a similar flavor as Theorem 1 for the parabolic analogue of (1) were obtained in [17] on bounded domains with Dirichlet boundary conditions. In that setting it was shown that solutions with initial data with asymptotically large, albeit dilute, numbers of vortices converge weakly to a mean field model, predicted by [12].…”

Well-posedness for mean-field evolutions arising in superconductivity

“…The first rigorous deductions of such (macroscopic) mean-field limit models from the (mesoscopic) 2D GinzburgLandau equation are due to [35,30,48]. As discovered by Serfaty [48], in the dissipative case α > 0, the limiting equation (1.6) is only correct in a regime of dilute vortices, while for a higher vortex density it must be replaced by the following compressible flow,…”

Gross--Pitaevskii Vortex Motion with Critically Scaled Inhomogeneities

“…Proof. Although it is possible to get explicit control on the error o ε (1) in (2.1) by a careful analysis as in [20,25], the weaker estimate (2.1) is sufficient to prove the vortex motion law. 1.…”