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

Abstract

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.

Finite dimensional global and exponential attractors for a coupled time-dependent Ginzburg-Landau equations for atomic Fermi gases near the BCS-BEC crossover

“…First, we confine the dynamics of system (1.3)-(1.5) to the regular set B1 ⊂ (H 2 ∩ H 1 0 ) × (H 2 ∩ H 1 0 ) obtained in Proposition 4.2. In order to prove the existence of an exponential attractor, we shall use the simple constructive method introduced in [8, Proposition 1] and follow the strategy in [2] (cf. also [12]).…”

“…We refer to [16] for a survey. In this paper, we apply a simple method that also works in Banach spaces, due to [8] (see [2,9,13] for generalizations) to prove the existence of an exponential attractor. As a byproduct, we obtain the finite fractal dimensionality of the global attractor.…”

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations

“…We refer to [16] for a survey. In this paper, we apply a simple method that also works in Banach spaces, due to [8] (see [2,9,13] for generalizations) to prove the existence of an exponential attractor. As a byproduct, we obtain the finite fractal dimensionality of the global attractor.…”

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

“…which can be proved in the same way as in superconductivity ( [8]), provided that this inequality holds at the initial instant. 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)…”

“…which can be proved in the same way as in superconductivity ( [8]), provided that this inequality holds at the initial instant. 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]).…”