Existence of superconducting solutions for a reduced Ginzburg-Landau model in the presence of strong electric currents

TL;DR

This paper proves the existence of superconducting solutions in a simplified Ginzburg-Landau model with strong electric currents, showing solutions are one-dimensional near boundaries and stable in a reduced form.

Contribution

It demonstrates the existence of solutions under strong currents in a reduced model, extending previous work by considering higher current densities and boundary behaviors.

Findings

Existence of solutions via non-convex minimization

Solutions are essentially one-dimensional near boundaries

Linear stability results for a simplified model

Abstract

In this work we consider a reduced Ginzburg-Landau model in which the magnetic field is neglected and the magnitude of the current density is significantly stronger than that considered in a recent work by the same authors. We prove the existence of a solution which can be obtained by solving a non-convex minimization problem away from the boundary of the domain. Near the boundary, we show that this solution is essentially one-dimensional. We also establish some linear stability results for a simplified, one-dimensional version of the original problem.