The 3D nematic liquid crystal equations with blow-up criteria in terms of pressure. (April 2018)

• Record Type: Journal Article
• Authors: Liu, Qiao; Wang, Pei
• Is Part Of: Nonlinear analysis. Volume 40(2018)
• Page Start: 290
• Page End: 306
• Publication Date: 2018-04
• Subjects: Nematic liquid crystal flows -- Local in time smooth solution -- Blow-up criteria -- Anisotropic Lebesgue spaces; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2017.08.008
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper, we concern the 3D nematic liquid crystal equations and prove three almost Serrin-type blow-up criteria for the breakdown of local in time smooth solutions in terms of pressure and gradient of the orientation field. More precisely, let T ∗ be the maximal time of the local smooth solution, then T ∗ < + ∞ if and only if ∫ 0 T ∗ ‖ ‖ ‖ P ( ⋅, t ) ‖ L x 1 p ‖ L x 2 q ‖ L x 3 r β + ‖∇d(⋅, t)‖ L 4 8 d t = ∞, with 2 β + 1 p + 1 q + 1 r = 2 and 2 ≤ p, q, r ≤ ∞, 1 − ( 1 p + 1 q + 1 r ) ≥ 0, and ∫ 0 T ∗ ‖ ‖ ‖ ∇ P ( ⋅, t ) ‖ L x 1 p ‖ L x 2 q ‖ L x 3 r β + ‖ ∇ d ( ⋅, t ) ‖ L 4 8 d t = ∞, with 2 β + 1 p + 1 q + 1 r = 3 and 1 ≤ p, q, r ≤ ∞, 1 − ( 1 2 p + 1 2 q + 1 2 r ) ≥ 0, and ∫ 0 T ∗ ‖ ‖ ∂ 3 P ( ⋅, t ) ‖ L x 3 γ ‖ L x 1 x 2 α β + ‖ ∇ d ( ⋅, t ) ‖ L 4 8 d t = ∞, with 2 β + 1 γ + 2 α = k ∈ [ 2, 3 ) and 3 k ≤ γ ≤ α < 1 k − 2 .