ABSTRACT
There are some materials which exhibit a sharp rise in conductivity at temperatures of the order of 5oK and currents started in these metals persist for a long time. This is the essence of superconductivity which was discovered by Kamerlingh Onnes in 1911 (cf.[1], [2], [5], [6], [7], [15], [16], [17]). He observed that the electrical resistance of various metals such as mercury, lead and tin disappeared completely in a small temperature range at a critical temperature Tc which is characteristic of the material. The complete disappearance of resistance is most sensitively demonstrated by experiments with persistent currents in superconducting rings. In 1914 Kamerlingh Onnes discovered that the resistance of a superconduc-
tor could be restored to its value in the normal state by the application of a large magnetic field. About ten years later, Tuyn and Kamerlingh Onnes performed experiments on cylindrical specimens, with the axis along the direction of the applied field, and showed that the resistance increases rapidly in a very small field interval. The value Hc of H at which the jump in resistance occurs is termed threshold field. This value Hc is zero at T = Tc and increases as the T is lowered below Tc. In the first part of the paper we recall the London model of superconduc-
tivity, the traditional Ginzburg-Landau theory and the dynamical extension presented by Gor’kov and E´liashberg [11]. These models are able to describe the phase transition which occurs in a metal or alloy superconductor, when
the temperature is constant, but under the critical value Tc. In these hypotheses the material will pass from the normal to the superconductor state if the magnetic field is lowered under the threshold field Hc. In this paper we present a generalization of the Ginzburg-Landau theory which considers variable both the magnetic field and the temperature. Also this model describes the phenomenon of superconductivity as a second-order phase transition. The two phases are represented in the plane H − T by two regions divided by a parabola. The second part of the paper is devoted to the proof of existence and unique-
ness of the solutions of the nonisothermal Ginzburg-Landau equations. In a previous paper ([3]) we have shown the well posedness of the problem obtained by neglecting the magnetic field. In this paper, the existence and the uniqueness of the solutions of the nonisothermal Ginzburg-Landau equations are proved after formulating the problem by means of the classical state variables (ψ,A, φ) together with the temperature u = T/Tc. The existence of the weak solutions in a bounded time interval is established by applying the Galerkin’s technique. Then, by means of energy estimates we obtain the existence of global solutions in time. Finally, we prove further regularity and uniqueness of the solutions.
