Local Minimizers with Unbounded Vorticity for the 2D Ginzburg‐Landau Functional. Issue 9 (1st March 2022)

• Record Type: Journal Article
• Authors: Contreras, Andres; Jerrard, Robert L.
• Is Part Of: Communications on pure and applied mathematics. Volume 75:Issue 9(2022)
• Page Start: 1997
• Page End: 2032
• Publication Date: 2022-03-01
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.22043
• Languages: English
• ISSNs: 0010-3640

Abstract

Abstract: A central focus of Ginzburg‐Landau theory is the understanding and characterization of vortex configurations. On a bounded domain Ω ⊆ ℝ 2, global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit ϵ → 0, where ϵ > 0 is the inverse of the Ginzburg‐Landau parameter. A notable open problem is whether there are solutions of the Ginzburg‐Landau equation for any number of vortices below h e x ∣ Ω ∣ / 2 π, for external fields of up to superheating field strength. In this paper, we prove that there are constants K 1, α > 0 such that given natural numbers satisfying 1 ≤ N ≤ h e x 2 π Ω − h e x − 1 / 4, local minimizers of the Ginzburg‐Landau functional with this many vortices exist, for fields such that K 1 ≤ h e x ≤ 1 / ϵ α . Our strategy consists of combining: the minimization over a subset of configurations for which we can obtain a very precise localization of vortices; expansion of the energy in terms of a modified vortex interaction energy that allows for a reduction to a potential theory problem; and a quantitative vortex separation result for admissible configurations. Our results provide detailed information about the vorticity and refined asymptotics of the local minimizers that we construct. © 2021 Wiley Periodicals LLC.