Viscosity solutions of a degenerate parabolic-elliptic system arising in the mean-field theory of superconductivity
type: Item
instance of: scholarly article
title: Viscosity solutions of a degenerate parabolic-elliptic system arising in the mean-field theory of superconductivity
zbMATH Open document ID: 0927.35117
author: Charles M. Elliott
author: Barbara E. E. Stoth
author: Reiner Schätzle
published in: Archive for Rational Mechanics and Analysis
publication date: 1999-05-17T00:00:00Z
review text: The authors consider the mathematical model of the mean-field theory of superconducting vortices described by the degenerate parabolic-elliptic system \[ \Psi_t-| \nabla\Psi|\Biggl( \sigma\nabla \Biggl({\nabla\Psi\over |\nabla\Psi|}+ q- \Psi\Biggr)\Biggr)= 0\quad\text{in }\Omega\times ]0,T[, \] \[ -\Delta q+ \kappa_\Omega q= \kappa_\Omega\Psi\quad\text{in }\mathbb{R}^2\times ]0,T[, \] together with a homogeneous Neumann condition for \(\Psi\) on \(\partial\Omega\times ]0,T[\) and Neumann condition \(q_\infty\) for \(q\) at infinity. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) and \(\sigma\) is a nonnegative constant. The existence and uniqueness of the solution are proved. The dynamic equation is solved in the viscosity sense while the elliptic equation is solved in the weak sense. The comparison principle is proved for the dynamic system. For \(\sigma= 0\), it is proved that \((\omega, H)= (\nabla^\perp\Psi, \nabla^\perp q)\) is the local weak solution of the model of the mean-field theory of superconductivity. (Here \(H\) is a magnetic field and \(\omega\) is the averaged vorticity.) In addition, the stationary problem is investigated for \(\sigma= 0\) and domains \(\Omega\) constructed that allow for solutions with more than two superconducting phases.
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 82D55
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1287914
zbMATH Keywords: mean-field theory of superconducting vortices
zbMATH Keywords: degenerate parabolic-elliptic system
zbMATH Keywords: Neumann condition
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: comparison principle
reviewed by: Bronius Kvedaras
MaRDI profile type: Publication
full work available at URL: s002050050125
OpenAlex ID: W2135189083
DOI: 10.1007/S002050050125
description: scientific article; zbMATH DE number 1287914