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

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 epsilon$\varepsilon$. 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|log epsilon|${H_{c_1}}+ C \log {|\log \varepsilon |}$, the total vorticity remains bounded independently of epsilon$\varepsilon$, 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.