The 3D nematic liquid crystal equations with blow-up criteria in terms of pressure. (April 2018) - Liu, Qiao; Wang, Pei
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 .