Existence and uniform boundedness of strong solutions of the time-dependent Ginzburg-Landau equations of superconductivity

Abstract

The time-dependent Ginzburg-Landau equations of superconductivity with a time-dependent magnetic field $\mathbf{H}$ are discussed. We prove existence and uniqueness of weak and strong solutions with $H^1$-initial data. The result is obtained under the “$\phi =-\omega \left(\nabla \cdot \mathbf{A}\right)$” gauge with $\omega >0$. These solutions generate a dynamical process and are uniformly bounded in time.