Bounded vorticity for the 3D GinzburgLandau model and an isoflux problem
Abstract
We consider the full threedimensional GinzburgLandau 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 GinzburgLandau 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 [Román, C. On the First Critical Field in the Three Dimensional GinzburgLandau Model of Superconductivity. Commun. Math. Phys. 367, 317349 (2019). https://doi.org/10.1007/s0022001903306w] 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 threedimensional setting a twodimensional result of [Sandier, E., Serfaty, S. GinzburgLandau minimizers near the first critical field have bounded vorticity. Cal Var 17, 1728 (2003). https://doi.org/10.1007/s0052600201589]. We finish by showing an improved estimate on the value of $H_{c_1}$ in some specific simple geometries.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.06858
 Bibcode:
 2021arXiv211006858R
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 35Q56 (35J50 49K10 82D55)
 EPrint:
 45 pages, 4 figures