Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain

Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain

“…Thanks to Propositions 2 and 3, Lemma 2.2, Lemma 2.3 and Gagliardo-Nirenberg inequality we show (see [2] for a complete proof) that for N > 0 fixed large enough,…”

“…Since A − 1 2 0 D 1 is a bounded operator from L 2 into itself (see [2]), we infer from Cauchy-Schwarz inequality and Proposition 2 that…”

“…At this stage, we need the following helpful lemma which is a consequence of Proposition 3 (see [2] for a complete proof), Lemma 2.8. There exist T > 0 independent of the trajectory, as in Proposition 3, and C T > 0 depending only on γ, σ, f and T such that for all admissible pair (q, p) satisfying p ∈ max 4 3 , 2 1+2σ , 2 , the following estimate holds…”

“…In this paper we aim to investigate some technical results established in [2] (see Propositions 2 and 3 in Section 2 below) to prove that for a less regular external force f belonging to L 2 (Ω), we have the following improvement results over [2] (except the existence of a compact global attractor in H ) that states as follows Theorem 1.1. The equation (1) defines a dynamical system in H which possesses a global attractor that is a compact subset of H 2 (Ω) with finite fractal dimension.…”

“…It is worth pointing out that the equation (1) was studied by B. Alouini and O. Goubet in [2] where they have proved, using some powerful tools from harmonic analysis, that for a regular external force f that does not depends on t and which belongs to H , the appropriate dynamical system possesses a compact global attractor in H , bounded in H 3 . In this paper we aim to investigate some technical results established in [2] (see Propositions 2 and 3 in Section 2 below) to prove that for a less regular external force f belonging to L 2 (Ω), we have the following improvement results over [2] (except the existence of a compact global attractor in H ) that states as follows Theorem 1.1.…”