On the analysis of a non-isothermal model for superconductivity

We analyze a model for non-isothermal superconductivity, derived independently by G. Maugin, and K. Miya and S. A. Zhou. The model is described by a parabolic system based on the Time-Dependent Ginzburg–Landau (TDGL) equation, the Maxwell equations, and an energy equation such that the Clausius–Duhem inequality holds. The principal unknown fields are the complex valued Ginzburg–Landau order parameter (psi), the magnetic vector potential A, and the temperature T. A significant feature for this model is that it accounts for the interchange of thermal and electro-magnetic energies through Joule heating. The sensitivity of superconducting materials to thermal variations is a major obstacle to their applications. In practice one sees that time varying currents and magnetic fields generate thermal energy, producing ‘hot spots’ and increasing the temperature. This can result in suppressing superconductivity, i.e. (psi) tending to 0. Our principal result is that we exhibit this phenomenon. We prove that if the electro-magnetic energy of the superconductor is sufficiently large at time t = 0 then T will rise beyond the critical temperature, T_c, and (psi) to 0 as t – (infinity). Earlier analytic work investigating electro-dynamic effects in superconductivity centered on an isothermal model consisting of the TDGL equation and Maxwell equations. This setting takes into account just electro-magnetic effects and produces markedly different evolutions.