Bounded vorticity for the 3D Ginzburg-Landau model and an isoflux problem

TL;DR

This paper investigates the onset of vorticity in the 3D Ginzburg-Landau model of superconductivity near the first critical magnetic field, establishing bounds on vorticity and linking vortex concentration to an isoflux problem.

Contribution

It extends the understanding of vortex behavior in three dimensions near the first critical field, proving bounded vorticity under certain conditions and connecting it to an isoflux maximization problem.

Findings

Vorticity remains bounded below a specific magnetic field threshold.

Vortex lines concentrate near the maximizer of the isoflux problem.

Improved estimates for the first critical field in simple geometries.

Abstract

We consider the full three-dimensional Ginzburg-Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the "first critical field" $H_{c_1}$ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg-Landau parameter $\varepsilon$. This onset of vorticity is directly related to an "isoflux problem" on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [Rom\'an, C. On the First Critical Field in the Three Dimensional Ginzburg-Landau Model of Superconductivity. Commun. Math. Phys. 367, 317-349 (2019). https://doi.org/10.1007/s00220-019-03306-w] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below $H_{c_1}+ C \log |\log \varepsilon|$, the total vorticity remains bounded independently of $\varepsilon$, with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three-dimensional setting a two-dimensional result of [Sandier, E., Serfaty, S. Ginzburg-Landau minimizers near the first critical field have bounded vorticity. Cal Var 17, 17-28 (2003). https://doi.org/10.1007/s00526-002-0158-9]. We finish by showing an improved estimate on the value of $H_{c_1}$ in some specific simple geometries.