An equivalence relation for the Ginzburg-Landau equations of superconductivity
In the Ginzburg-Landau theory of phase transitions, the state of a superconductor is described by a complex-valued order parameter \(\psi\), a real vector-valued vector potential \(A\), and a real scalar-valued function \(\phi\), which has the signification of an electric potential. The main result of this paper gives a characterization of those stationary states in \({\mathcal W}^{1,2}\times L^2\) which are gauge equivalent with a solution \((\psi ,A,\phi )\) of the time-dependent Ginzburg-Landau equation. It is shown that these states must be of the form \((\psi_0,A_0,0)\), where \((\psi_0,A_0)\) is a solution of the time-independent Ginzburg-Landau equation. In the last section of the paper the authors analyze the time-dependent Ginzburg-Landau equation in the ``\(\phi =-\omega (\nabla \cdot A)\)'' gauge and it is proven that a weak solution of this equation tends to a weak solution of the time-independent Ginzburg-Landau equation in the London ``\(\nabla\cdot A=0\)'' gauge as the time goes to infinity, for all \(\omega >0\).
scientific article; zbMATH DE number 1057389