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 “Æ=−É(∇⋅A)†gauge with É>0. These solutions generate a dynamical process and are uniformly bounded in time
The problem of existence of time-periodic solutions for the time-dependent GinzburgLandau equations of superconductivity is discussed. It is assumed that the applied magnetic field H is τ -periodic in time, and so is the associated dynamical process. We prove the existence of τ -periodic solutions in time, which are exactly the fixed points of the associated period mapping.
Mathematics Subject Classification (2000). Primary 35K55. Secondary 35B40, 35B65, 35K15, 82D55.
Key words superconductivity, Ginzburg-Landau equations, initial boundary-value problems, periodic solutions, stability MSC (2000) Primary 35K55, 35B35. Secondary 35B10, 35B40, 35K15, 82D55The time-dependent Ginzburg-Landau equations describe the state of a superconducting material. The case of time-periodic applied magnetic field implies the existence of periodic orbits. Sufficient conditions are given here that, if satisfied, guarantee the stability of some associated periodic solutions.
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