Interacting helical vortex filaments in the three-dimensional Ginzburg–Landau equation
Entity
UAM. Departamento de MatemáticasPublisher
EMS PressDate
2019-07-31Citation
Journal of the European Mathematical Society 24. 12(2022): 4143-4199ISSN
1435-9855 (print); 1435-9863 (online)Funded by
J. Dávila has been supported by a Royal Society Wolfson Fellowship, UK. M. del Pino has been supported by a Royal Society Research Professorship, UK. M. Medina has been partially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 754446 and UGR Research and Knowledge Trans fer – Found Athenea3i, and by Project PDI2019-110712GB-100, MICINN, Spain. R. Rodiac has been partially supported by the F.R.S.–FNRS under the Mandat d’Impulsion scientifique F.4523.17, “Topological singularities of Sobolev maps” and by the ANR project BLADE Jr. ANR-18-CE40-0023. Part of this work was carried out during a visit of R. Rodiac at the Universidad de Chile and Pontificia Universidad Católica de Chile supported by the project REC05 Wallonia-Brussels/ChileProject
info:eu-repo/grantAgreement/EC/H2020/754446/EU//Marie Sklodowska-Curie; Gobierno de España. PDI2019-110712GB-100Subjects
Ginzburg–Landau equation; logarithmic n-body problem; renormalized energy; vortex filamnets; MatemáticasRights
© 2022 European Mathematical SocietyAbstract
For each given n ≥ 2, we construct a family of entire solutions u"(z, t), ε > 0, with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation ∆u + (1 - ⃒u⃒2)u = 0, (z, t) ∊ R2 × R ≃ R3. These solutions are 2π=ε-periodic in t and have n helix-vortex curves, with asymptotic behavior, as ε → 0, (Formular Presented) where W (z) = w(r)ei θ , z = r ei θ , is the standard degree +1 vortex solution of the planar Ginzburg–Landau equation ∆W + (1 - ⃒W ⃒2)W = 0 in R2 and (Formular Presented) Existence of these solutions was previously conjectured by del Pino and Kowalczyk (2008), f (t) =(f1(t), . . ., fn(t)) being a rotating equilibrium point for the renormalized energy of vortex filaments derived there,(Formular Presented) corresponding to that of a planar logarithmic n-body problem. The modulus of these solutions converges to 1 as ⃒z⃒ goes to infinity uniformly in t, and the solutions have nontrivial dependence on t, thus negatively answering the Ginzburg–Landau analogue of the Gibbons conjecture for the Allen–Cahn equation, a question originally formulated by H. Brezis. © 2022 European Mathematical Society Published by EMS Press
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Google Scholar:Dávila, Juan
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del Pino, Manuel
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Medina, María
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Rodiac, Rémy
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