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

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’ Hc1$H_{c_1}$ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg–Landau parameter ε$\varepsilon$ . 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 [22] 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 Hc1+Clog|logε|${H_{c_1}}+ C \log {|\log \varepsilon |}$ , the total vorticity remains bounded independently of ε$\varepsilon$ , with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three‐dimensional setting a two‐dimensional result of [28]. We finish by showing an improved estimate on the value of Hc1${H_{c_1}}$ in some specific simple geometries.