Local minimizers with unbounded vorticity for the $2$d Ginzburg-Landau functional

A central focus of Ginzburg-Landau theory is the understanding and characterization of vortex configurations. On a bounded domain $\Omega\subseteq $\mathbb{R}$^2,$ global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit $ϵ\to 0,$ where $ϵ>0$ is the inverse of the Ginzburg-Landau parameter. The presence of an applied magnetic field of strength $h_{ex}\gg 1$ makes possible the existence of stable vortex states. A notable open problem is whether there are solutions of the Ginzburg-Landau equation for any number of vortices below $h_{ex}\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{/}2\pi ,$ for external fields of up to super-heating field strength. The best earlier partial results give, for every $0<c<1,$ and $K>0,$ the existence of local minimizers of the Ginzburg-Landau functional with a prescribed number of vortices in the range $1 \leq N \leq \min { K | \log \epsilon |, c ( h_{ex} |\Omega| /2 \pi ) }$andforvaluesof$1\ll_\epsilon h_{ex}$ smaller than a power of the Ginzburg-Landau parameter. In this paper, we prove that there are constants $K_1,\alpha >0$ such that given natural numbers satisfying $$1\le N\le \frac{h_{ex}}{2\pi }(\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }-h_{ex}^{-1\mathrm{/}4}),$$ local minimizers of the Ginzburg-Landau functional with this many vortices exist, for fields such that $K_1\le h_{ex}\le 1\mathrm{/}{ϵ}^{\alpha }\mathrm{.}$ Our strategy consists in combining: the minimization over a subset of configurations for which we can obtain a very precise localization of vortices; expansion of the energy in terms of a modified vortex interaction energy that allows for a reduction to a potential theory problem; and a quantitative vortex separation result for admissible configurations. Our results provide detailed information about the vorticity and refined asymptotics of the local minimizers that we construct.