@article{34b39d94e7b64e8e8984ef88e58c1fb3,

title = "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 {\textquoteleft}first critical field{\textquoteright} (Formula presented.) at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg–Landau parameter (Formula presented.). This onset of vorticity is directly related to an {\textquoteleft}isoflux problem{\textquoteright} 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 (Formula presented.), the total vorticity remains bounded independently of (Formula presented.), 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 (Formula presented.) in some specific simple geometries.",

author = "Carlos Rom{\'a}n and Etienne Sandier and Sylvia Serfaty",

note = "Funding Information: Carlos Rom{\'a}n acknowledges funding from the Chilean National Agency for Research and Development (ANID) through FONDECYT Iniciaci{\'o}n Grant 11190130. He wishes to thank the support and kind hospitality of the Courant Institute of Mathematical Sciences and the Paris‐Est University, where part of this work was done. Etienne Sandier wishes to thank the kind hospitality of the Max Planck Institute for Mathematics in the Sciences in Leipzig, where part of this work was done. Sylvia Serfaty acknowledges funding from NSF Grant DMS‐2000205 and the Simons Investigator program. Finally, the authors wish to thank the anonymous referee for numerous useful comments which helped improve this article. Funding Information: Carlos Rom{\'a}n acknowledges funding from the Chilean National Agency for Research and Development (ANID) through FONDECYT Iniciaci{\'o}n Grant 11190130. He wishes to thank the support and kind hospitality of the Courant Institute of Mathematical Sciences and the Paris-Est University, where part of this work was done. Etienne Sandier wishes to thank the kind hospitality of the Max Planck Institute for Mathematics in the Sciences in Leipzig, where part of this work was done. Sylvia Serfaty acknowledges funding from NSF Grant DMS-2000205 and the Simons Investigator program. Finally, the authors wish to thank the anonymous referee for numerous useful comments which helped improve this article. Publisher Copyright: {\textcopyright} 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.",

year = "2023",

month = mar,

doi = "10.1112/plms.12505",

language = "English (US)",

volume = "126",

pages = "1015--1062",

journal = "Proceedings of the London Mathematical Society",

issn = "0024-6115",

publisher = "Oxford University Press",

number = "3",

}