2016/06/02 by Contreras, Andres, Jerrard, Robert L. · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1606.00732
We introduce a framework to study the occurrence of vortex filament concentration in 3D Ginzburg-Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical 3-dimensional domain, in certain scaling limits; it is shown to arise as the Γ-limit of a sequence of scaled Ginzburg-Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg-Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically relevant situations where clustering of filaments is expected. In this setting it also constitutes a higher-order asymptotic expansion of the Ginzburg-Landau energy, a refinement over the arclength functional approximation. Our description of the vorticity region significantly improves on previous studies and enables us to rigorously distinguish a collection of multiplicity one vortex filaments from an ensemble of fewer higher multiplicity ones. As an application, we prove the existence of solutions of the Ginzburg-Landau equation that exhibit clusters of vortex filaments whose small-scale structure is governed by the limiting free-energy functional.