Asymptotic behaviour of time‐dependent Ginzburg–Landau equations of superconductivity

Existence and uniform boundedness of strong solutions ofthe time‐dependent Ginzburg‐Landau equations of superconductivity

“…Inequality (5.41) confirms the uniqueness of the solutions with respect to initial data. Furthermore, Theorem 5.9 extends the result in [15] (cf. Theorem 3.1) to the nonstationary case.…”

Time-periodic solutions of the time-dependent Ginzburg-Landau equations of superconductivity

“…Various properties of the solutions for system (1.1)-(1.4) have been investigated by many authors; see [3], [9], [10], [11], [12], [15] and [16]. We restrict ourselves to the three dimensional case, because the discussion is similar for the case of n = 2 .…”

Global and exponential attractors for a Ginzburg-Landau model of superfluidity

“…In the context of superconductivity, the same problem has been treated in [21], where the authors prove existence of the global attractor. Later, Rodriguez-Bernal et al [20] show that the semigroup generated by the system admits finite-dimensional exponential attractors. The main difference and difficulty in our problem is due to the presence of the absolute temperature which does not appear in the traditional Ginzburg-Landau equations of superconductivity, where an isothermal model is analyzed.…”

“…Relation (1.4) is widely exploited in [3] and [5] to prove that the Ginzburg-Landau system of superconductivity admits absorbing sets, global and exponential attractors. As a matter of facts the inequality (1.4) is not used neither in [20] nor in [21], where existence of the global attractor is proved by means of a Lyapunov functional and exponential attractors are obtained as a consequence of the squeezing property of the solutions ( [9]). Therefore, in this paper we construct a Lyapunov functional for system (1.1)-(1.3) which allows to show existence of the global attractor consisting of the unstable manifold of the stationary solutions.…”