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

Abstract

The time-dependent Ginzburg-Landau equations of superconductivity with a time-dependent magnetic field H are discussed. We prove existence and uniqueness of weak and strong solutions with H1-initial data. The result is obtained under the “φ=−ω(∇⋅A)” gauge with ω>0. These solutions generate a dynamical process and are uniformly bounded in time

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

“…The proofs of the theorems are given in the next section. We first need to recall some results from our earlier article [16]. Assume that Ω is a bounded domain in R n (n = 2 or 3) , with boundary ∂Ω of class C 1,1 and γ : ∂Ω −→ R is a nonnegative Lipschitz continuous function.…”

“…On the other hand according to [16,Theorem 3], the solutions of (3.10) are bounded for all t ≥ 0 , and if ψ 0 ∞ ≤ 1 it yields the following estimate…”

“…First we start by showing Theorem 1, where we use an estimate on the solutions of the TDGL equations established in [16]. In what follows C will denote various constants independent on t and depending only on the data κ, η, H and the constants entering equations (2.1)-(2.3).…”

“…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 .…”

“…We restrict ourselves to the three dimensional case, because the discussion is similar for the case of n = 2 . The existence, uniqueness, and dynamical properties of solutions of the TDGL equations, when the magnetic field H is time-dependent and in some gauge equivalence classes, were obtained in [3], [9] and quite recently in [16]. It was shown in [3] that the TDGL equations define a dynamical process, whose asymptotic behavior depends on the nature of the magnetic field H .…”