Convergence to equilibrium for parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations. (English) Zbl 1183.35258

Grasselli, Maurizio; Wu, Hao; Zheng, Songmu

SIAM J. Math. Anal. 40, No. 5, 2007-2033 (2009).

A Ginzburg-Landau-Maxwell model of superconductivity is considered in a bounded, 2-dimensional domain \(\Omega\) with smooth boundary \(\partial\Omega\) and the large-time asymptotic behaviour of solutions is investigated. Under the choice of Coulomb gauge the magnetic potential \(A\) and electric potential \(\Phi\) are used to describe the electromagnetic field. In this model a complex-valued function \(\psi\) describes the state of superconductivity, with \(\left|\psi\right|^{2}\) representing the concentration of superconducting electrons. The dynamics of \(\psi\) is coupled with the magnetic potential \(A\) and the dynamics of the potentials \(\Phi\), \(A\) is coupled with \(\psi\), leading to a hyperbolic-parabolic coupling. The main concern of the paper is to show that solutions tend to the equilibrium state as time goes to infinity. A power law decay estimate is given for this convergence to a unique equilibrium solution.

Reviewer: Rainer Picard (Dresden)

MSC:

35Q56	Ginzburg-Landau equations
35B40	Asymptotic behavior of solutions to PDEs
82D55	Statistical mechanical studies of superconductors
78A25	Electromagnetic theory, general

Keywords:

Ginzburg-Landau-Maxwell equations; asymptotic behavior; superconductivity; Coulomb gauge; convergence to equilibrium; Lojasiewicz-Simon inequality