Asymptotic behaviour of time-dependent Ginzburg-Landau equations of superconductivity

Rodriguez-Bernal, Anı́bal; Wang, Bixiang; Willie, Robert

doi:10.1002/(sici)1099-1476(199912)22:18<1647::aid-mma97>3.0.co;2-w

Cited by 11 publications

(4 citation statements)

References 15 publications

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“…For detailed physical description and mathematical modeling of the superconductivity phenomena, we refer to the review articles [10,20]. Theoretical analysis for the TDGL equations can be found in literature [14,21,29,38,39]. The global existence and uniqueness of the strong solution were established in [14] for the TDGL equations with the Lorentz gauge.…”

Section: Introductionmentioning

confidence: 99%

Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity

Gao

Sun

2017

Adv Comput Math

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A linearized backward Euler Galerkin-mixed finite element method is investigated for the time-dependent Ginzburg-Landau (TDGL) equations under the Lorentz gauge. By introducing the induced magnetic field σ = curl A as a new variable, the Galerkin-mixed FE scheme offers many advantages over conventional Lagrange type Galerkin FEMs. An optimal error estimate for the linearized Galerkin-mixed FE scheme is established unconditionally. Analysis is given under more general assumptions for the regularity of the solution of the TDGL equations, which includes the problem in two-dimensional noncovex polygons and certain three dimensional polyhedrons, while the conventional Galerkin FEMs may not converge to a true solution in these cases. Numerical examples in both two and three dimensional spaces are presented to confirm our theoretical analysis. Numerical results show clearly the efficiency of the mixed method, particularly for problems on nonconvex domains.

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Section: Introductionmentioning

confidence: 99%

Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity

Gao

Sun

2017

Adv Comput Math

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Berti¹,

Berti²,

Bochicchio³

2011

Discrete &Amp; Continuous Dynamical Systems - S

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The long-time behavior of the solutions for a non-isothermal model in superuidity is investigated. The model describes the transition between the normal and the superuid phase in liquid 4He by means of a non-linear di erential system, where the concentration of the superuid phase satisfi a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techinques of semigroups theory, we prove the existence of an exponential attractor offnite fractal dimension which contains the global attractor

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“…However, no asymptotic result seems to have appeared in the literature, even if the longtime behavior of the solutions is investigated for the quasi-steady model (cf. [9,10]). In both papers, the superconductor occupies a bounded domain Ω ⊂ R 2 and the phasespace is L 2 (Ω) × L 2 (Ω).…”

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confidence: 99%

Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations

Berti

Gatti

2006

Quart. Appl. Math.

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Abstract. This article is devoted to the long-term dynamics of a parabolic-hyperbolic system arising in superconductivity. In the literature, the existence and uniqueness of the solution have been investigated but, to our knowledge, no asymptotic result is available. For the bidimensional model we prove that the system generates a dissipative semigroup in a proper phase-space where it possesses a (regular) global attractor. Then, we show the existence of an exponential attractor whose basin of attraction coincides with the whole phase-space. Thus, in particular, this exponential attractor contains the global attractor which, as a consequence, is of finite fractal dimension.

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Asymptotic behaviour of time-dependent Ginzburg-Landau equations of superconductivity

Cited by 11 publications

References 15 publications

Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity

Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity

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

Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations

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